A-i9i2 EVIEWI OF MEtHODS FOR GENERATIN G SVNTHETIC SEISMOGRRM5 /

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REPORT85-10US Army Corpsof EngineersCold Regions Research &Engineering Laboratory

Review of methods for generatingsynthetic seismograms

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CRREL Report 85-10June 1985

Review of methods for generating 1

synthetic seismograms

Lindamae Peck

A 4

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Prepared forOFFICE OF THE CHIEF OF ENGINEERS

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Approved for public release; distributlor Is unlimited.

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Unclassified •IS SECURITY CLASSIFICATION OF THIS PAGE CUMN an DAtE Entered) R O NE

- .RPORT OCUMETATIO PAGEREAD INSTRUCTIONS". -REOR DOUETTO PAG BEFORE COMPLETING FORM'.-]

1. REPORT NUMBER 2. GOVT ACCESSION No. 3. RECIPIENT'S CATALOG NUMBER

CRREL Report 85-10 9 ~Y4. TITLE (and Subtitle) S. TYPE OF REPORT & PERIOD COVERED

REVIEW OF METHODS FOR GENERATINGSYNTHETIC SEISMOGRAMS

6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(s) S. CONTRACT OR GRANT NUMBER(o)

Lindamae Peck

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASK

AREA & WORK UNIT NUMBERSU.S. Army Cold Regions Research and Engineering Laboratory DA Project 4AI6I I02AT24Hanover, New Hampshire 03755-1290 Task B, Work Unit 005

I1. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

Office of the Chief of Engineers June 1985Washington, D.C. 20314 13. NUMBER OF PAGES48 i(

14. MONITORING AGENCY NAME & ADDRESS(II different from Controlling Office) 15. SECURITY CLASS. (of thie report)

Unclassified...

15a. DECLASSI FICATION/ DOWNGRADINGSCHEDULE '

16. DISTRIBUTION STATEMENT (of this Report)

Approved for public release; distribution is unlimited.

17. DISTRIBUTION STATEMENT (of the abstract entered In Block 20, If different from Report)

IU

IS. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on revere* side It necessary end identify by block number)

Geophysics SeismologyGround motion Synthetic seismograms- - 'Mathematical analysisSeismic signatures

20. ABrRAcr (Omthue -t revers eI fH ne..amy anzd Identify by block num~ber)Various methods of generating synthetic seismograms are reviewed and examples of recent applicatiors of the methodsare cited. Body waves, surface waves, and normal modes are considered. The analytical methods reviewed include geo-metric ray theory, generalized ray theory (Cagniard-de Hoop method), asymptotic ray theory, reflectivity method, fullwave theory, and hybrid methods combining ray theory and mode theory. Two numerical methods, those of finite dif-ferences and finite elements, and a hybrid method combining finite differences with asymptotic ray theory are describedLimitations on the application or validity of the various methods are stated.

DD FO 1473 EDITION OF I NOV 6S IS OBSOLETE UnclassifiedSECURITY CLASSIFICATION OF THIS PAGE (ltbn Date Entered)

PREFACE

This report was prepared by Dr. Lindamae Peck, Geophysicist, of the Geophysical SciencesBranch, Research Division, U.S. Army Cold Regions Research and Engineering Laboratory. Fund-ing for this research was provided by DA Project 4A161102AT24, Research in Snow, Ice, and

Frozen Ground, Task B, Cold Regions Environmental Interactions, Work Unit 005, ComputationalMethods fir Seismic Wave Propagation in Cold Regions.

The author thanks Donald G. Albert and Dr. Jerome B. Johnson for technically reviewing themanuscript of this report.

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-'-" CONTENT'S PagPage

A bstract ......................................................................................................... . . .... . . IPreface ............................................................................................................................... iNomenclature ...................................................... ivSection 1. Introduction ................................................................................................... 1Section 2. Wave propagation in the earth ........................................................................ 2Section 3. Body waves: ray theory and wave theory ..................................................... 6 -

,.- G eom etric ray theory ................................................................................................... . 6Wave theory ..................................................... 10

Section 4 . Surface w aves ................................................................................................. 14 - -,

Section 5. N orm al m odes ................................................................................................ 17 , "-Section 6. Finite-difference method ................................................................................ 20Section 7. Finite-element method ................................................................................... 28Section 8. Hybrid m ethods ............................................................................................... 33Section 9. Conclusion ...................................................................................................... 34L iterature cited ................................................................................................................... 36

ILLUSTRATIONS

Figure1. Wave fronts shown in two dimensions .................................................................... 32. Wave fronts of a plane wave .................................................................................... 43. Wave types in bounded and unbounded media ....................................................... 54. Ray for each reflected plane wave and each transmitted plane wave generated

by a plane wave incident at an angle at a planar boundary ........................... 55. Reflected waves and transmitted waves generated by plane waves incident at

the w aveguide boundaries ............................................................................... 66. Rays A and B emanating from a source S at takeoff angles .................................. 77. Dispersion curves for three modes............................................................. 168. Geologic model for which the dispersion equation for Love waves is calculated ... 169. Graphic solution of eq 12 for the dispersion of Love waves in a single layer

over a half-space .............................................................................................. 1610. Wave fronts impinging on boundaries of waveguide at angle y ............................. 17I1. Plane wave representation of normal modes of the first three orders .................. 1912. Grids and templates for finite-difference method ................................................... 2113. Application of the finite-difference method ........................................................... 2214. Forces Q and displacements 6 at nodes of a quadrilateral element ....................... 2915. Calculated values of ground motion due to movement along a right-lateral fault 31

TABLES

Table1. Applications of the finite-difference method to seismology ................................... 272. Comparison of normal-mode and ray theories ........................................................ 333. Summary of seismic interpretation techniques ........................................................ 344. Characteristics of methods of generating synthetic seismograms ............................ 35

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NOMENCLATURE

a, b coefficientsc velocity; phase velocityf dimensionless imaginary number in wave potential for refracted wave when

angle of refraction is complex; function; frequency

1 ~VT2ik wave number =IPr(t) source term in full wave theoryn index of refraction; mode number 0p ray parameter = sinO/c; time level in finite difference scheme

SuejAAp E) 1 2-., e2

r range,= or = AX2+y2 z; radial distance from the center of the earthre radius of the earthS imaginary parameter analogous to wave frequency !At timeU. Wt displacementuMI notation for displacement at a point (mAx, nat) at a time pAt in finite differ-

ence schemex, J, z Cartesian spatial coordinates

4 Ax, Az, At spacing between points in a space-time grid in finite difference schemeA, B amplitudesC velocity; amplitudeC(p) product of all reflection and transmission coefficients for a rayCI velocity of Rayleigh wave at the surface of a half-space

ICI damping matrix in finite element schemeE amount of energy per unit time per unit solid angleF(w) source time functionG Green's functionH Heaviside function; width of a waveguide or thickness of a layerHo Hankel function

Iintensity of wave (amount of energy per second crossing a unit area)K 0 modified Bessel function[K], [A], [G] stiffness matrix or related matrix in finite-element schemeM amplitude[MI inertia matrix in finite-element schemeP pressure fieldIPI [time-independent, complex force amplitude

column vector representing external forces applied to nodal points of a finiteelement grid .)--

R /r- +Z2 ; reflection coefficientR reflectivity (composite reflection coefficient) SS coordinate distance measured along a ray; amplitudeT period of wave; phase function; transmission losses; travel timeU group velocity; displacementV velocityW wave frontX, Z Cartesian spatial coordinates

iv

" - " " . - " -" " -" : . " -" " - - . . . . - " .. . . . . , . .. . 5 ,S

a compressional wave velocity0 shear wave velocity'N Lamd constant; wavelength/ Lam6 constant (rigidity)'"" ?

a X+2Ma

p densityCO angular frequency =--- = Vk

A angular separation between source and receiver measured from center of earth "' -r travel time of a ray; phase shiftT(p) T(p)-pX(p)0 angle of incidence or angle of emergence .0 (P) PrzA+7(P)8 imaginary part of complex angle of refraction; displacement161 column vector representing displacement field of nodal points in finite-element grid7 parameter analogous to radial wave number; angle of incidence

2 2o-, 1 - , - kcsin yA(t) time series in full wave theory

7 a [( 1/C,2 ) _ p I ] /27 .

e arbitrary radius¢, 1 wave potential

wave potential

r [Mx)+2p(x)] (au/ax), P-waves or /(x) (au/ax), S-wavesir 3.i4159...• convolution

In imaginary part of a complex number or functionRe real part of a complex number or function

0"-°

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* REVIEW OF METHODS FOR GENERATING

SYNTHETIC SEISMOGRAMS

Lindamae Peck

SECTION 1. INTRODUCTION

A seismogram is a time record of ground motion. Displacements from the rest position of therecording instrument correlate with the arrival of various seismic waves at the location of the in-strument, although identification of the waves and determination of precise arrival times may bedifficult because of the presence of noise on the seismogram. The characteristics of a seismogram(wave arrivals, time between arrivals, amplitude and direction of displacements) are determinedby the nature of the source, the structure of the earth along the wave path, and the characteristicsI of the receiver (recording instrument). The equivalent information is required to create a synthet-ic seismogram: a mathematical representation of the source, a model of the geology betweensource and receiver, and a surface position designated as the location of the receiver, which ismodeled mathematically in terms of its response characteristics. The model of subsurface geologyis usually simplified to represent gross structural features and ignore details of the geology (ifknown) in the vicinity of the source and receiver. The characteristics of a synthetic seismogramare most sensitive to features (velocity and density profiles, layers, irregular structure) of the geo-logical model employed.

The usefulness of a synthetic seismogramn lies in how closely it duplicates actual seismograms.The degree of correlation between the two is a measure of the accuracy of the geological modelused in computing the synthetic seismogram. When a structure on the scale of the Earth itself ismodeled, an Earth model that results in a close match between synthetic and actual seismogramsis unlikely to be unique. Usually, however, a selection among the models can be made based uponindependent criteria such as consistency with petrological or thermodynamic models of the Earth'sinterior. When structure on the scale of a few kilometers or less is modeled, the problem of theuniqueness of the model is less and the model is often sufficiently well constrained on the basisof surface geology or seismic surveys.

Synthetic seismograms are constructed for several purposes. They are useful in investigationsof the geology of an area. The correlation between recorded and synthetic seismograms aids inthe construction and refinement of a geologic model of the area. Based on a preliminary model soobtained, the deployment of seismic surveys to resolve the subsurface structure can be done ef-ficiently and relatively economically. In this way, a synthetic seismogram is useful as a recoll-naissance tool and provides information necessary for engineering site studies. Other applications

kF 10 U sPA G

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,--2

of synthetic seismograms require a previous knowledge of the geology. When a sufficiently accu-rate geologic model is known, synthetic seismograms can be used to predict the effects of explo-sions or earthquakes (the amplitude of the ground motion at a site due to a source of given mag-nitude) or to derive a theoretical model of vehicle-generated or artillery-generated ground motionfor use in detection and classifier design.

The validity of a synthetic seismogram is dependent upon the accuracy of the geologic modelemployed, the applicability of the assumptions inherent in the derivation of the wave equation,and the approximations incorporated in the solution to the wave equation.

The wave equation is derived from a statement of the balance of forces acting on a unit volumeof unbounded medium (e.g. Grant and West 1965, Officer 1974, Aki and Richards 1980). Severalassumptions generally are made that result in a wave equation of the form O

a2 2a

2 a2 a2ax

where t is time, c is velocity, and x,y and z are spatial (Cartesian) coordinates. One assumption inderiving this equation is that external forces such as gravity acting on the medium may be neglect-ed. A second is that the medium is linearly elastic; this determines the form of the stress-strainrelation that is substituted in the balance of force equation. A third is that the medium is hono-geneous and isotropic and therefore the elastic parameters (e.g. the Lam6 constants) of the mediumare not functions of the spatial coordinates.

A synthetic seismogram is generated by solving the wave equation subject to the constraintthat the boundary conditions of the problem are satisfied. The boundary conditions pertain tothe components of stress and displacement at the free surface of the medium, at any interfaceswithin the medium, and at infinite depth in the medium. Various analytic and numerical methodsare available for use in solving the wave equation; a distinction among them is that, in general,analytic methods yield an approximate solution to the wave equation while numerical methods pyield an exact solution to an approximation of the wave equation. The selection of a particularmethod is based upon the wave types that are to be represented in the seismogram and on the de-gree of structural complexity that the method of generating the seismograms is capable of han-dling.

This report presents a description and evaluation of commonly used methods of generatingsynthetic seismograms. Its intent is to indicate the strengths and weaknesses of the methods andto provide examples of recent applications of them. The cited literature serves as an initial refer-ence for individuals to use in pursuing an approach to a particular problem. A concern of this re-port is to identify methods capable of generating synthetic seismograms for physical situationssuch as thin beds or near-surface velocity inversions that are typical of winter seismological condi-tions. The mathematical representation of source characteristics and receiver response is not con-sidered here- the cited references and general texts such as that of Aki and Richards (1980) showhow such factors are incorporated in the process of generating synthetic seismograms.

SECTION 2. WAVE PROPAGATION IN THE EARTHS

The types of seismic waves are distinguished by particle motion and velocity of propagation.The compressional wave (or synonymously the irrotational wave, longitudinal wave, dilatationalwave or P-wave) propagates at a velocity of ot = N(X+2p)/p through an isotropic, homogeneousmedium that is alternately condensed and rarefied; the particle motion is parallel to the directionof wave propagation. Here, X and p are the Lamd constants and p is the density of the medium.The shear wave (or synonymously the rotational wave, transverse wave or S-wave) propagates at

2. '

r r~~ - ----- - -

a velocity/3 ../pp distortion of the medium is in the plane perpendicular to the direction ofpropagation and is designated as SH (horizontal) or SV(vertical) depending on its orientation.

The P- and S-waves are body waves.Rayleigh waves are coupled P-SV waves with motion that at the surface defines an ellipse per-

pendicular to the surface and that exponentially decays in amplitude with depth; the waves aredispersive (propagation velocity depends on frequency) in a layered medium and nondispersive(with a velocity Cr equal to 0.919403 when Poisson's ratio is ) when propagating at the surfaceof a half-space of shear velocity 0. Lo~c waves are Sl waves that propagate sinusoidaly within alayer, decay exponentially with depth outside the layer, and exhibit dispersion; the velocity ofthe waves approaches the shear velocity of the surface layer in the limit of high frequencies and ...

approaches the (higher) shear velocity of the underlying half-space in the limit of low frequencies.

Rayleigh waves and Love waves are called surface waves. They are generated along with body - "'"waves (unless the source is located at a depth such that surface waves are not excited) and propa- - -

gate horizontally at the surface with an attenuation due to geometrical spreading that is propor- - -

tional to 1/(range) -. Since body waves are attenuated as 1r and head waves (see below) as l1/r 2 ,

the major portion of the wave energy reaching points distant from the source is carried in surface

waves. Surface waves are viewed as noise to be eliminated when vertically or near vertically inci-

dent (reflected) waves are of interest, as is typically the case in exploration seismology, or as use-ful counterparts to body waves when analyzed to extract information on source parameters and

earth structure between source and receiver. Because of the conditions under which they exist

and the manner in which they propagate, surface waves are exceptionally well suited for use inthe delineation of near-surface layered structure.

Diffracted waves are present when plane waves are incident on a feature that has a radius ofcurvature less than or equal to the wavelength of the incident waves. A discontinuous boundary - "(such as a geologic bed displaced by a fault) or a nonplanar boundary commonly causes diffracted

waves to be present in the wavefield. At distances from the diffracting source that are large rela-tive to wavelength, Huygens' construction (e.g. Halliday and Resnick 1974) is used to determine

the diffracted wave energy at a particular location.A point source in a homogeneous, isotropic medium generates P-waves (and S-waves if the

medium is nonfluid) that radiate in spherical fronts; i.e. the waves propagate in all directionsfrom the source and points of the medium that are in the same phase of motion are located on

surfaces (wave fronts) that define spheres. As a wave propagates away from the source, the radiusof curvature of its wave fronts increases and the wave fronts become increasingly planar over a

limited region (e.g. Halliday and Resnick 1974, Fig. 1). The wave fronts of a plane wave are paral-lel planes perpendicular to the direction of propagation of the wave; the rays are parallel lines nor-

mal to the wave fronts. Figure 2 shows wave fronts and rays of a plane wave propagating in thex-z plane at a velocity V; the propagation direction of the plane wave is at an angle 0 to the z-axis. .

Figure 1. Wave fronts shown in two/ / ( I ' " dimensions as dashed lines emanating

\ \ / from the point source, S. Radial/ arrows normal to the wave fronts are

-\.\_i / / / rays. Each ray indicates the direction/ of propagation of the wave. The spheri-

A i / cal wave fronts become locally planaras distance from S increases.

3

.' "1

. . . . . . ..'. -"-" -"-"-- - - - 7- - - - - --"-" .- - - - - --"-"- --. .•. .;.•. .."."'" ;'"""" - -" . ..".-.--.. .-. .,-.-. .-.-....... . .'-,-.--.. .-.-.. .

/ / ×ure 2. Ware frnts oa plane ware shown

/ as dashed lines. Rays art shown as arrows.

The plane wave may be represented by the potential

S-1 exp i(A x + kA - WO) (1)

wAhere .I is the amplitude of the wave potential: k, k sinO and k, = k cosO are the horizontal and 71vertical components of the wave number (where A- 27r/N and X is wavelength), c = 27r/T= Vk isangular frequency and 7 is period.

In a homogeneous, isotropic, unbounded medium, the only waves that arrive at the receivert1o11 anl iot rO)pic solrce are direct waves (Fig. 3a). In a homogeneous, isotropic half-space, surface-

reflected waves (and Rayleigh waves) also arrive at the receiver (Fig. 3b). If velocities (a, 0) in-,_roas t lto ril, with depth in the halt-space, refracted waves arrive at the receiver after travelingmininmum-tinte paths (Fig. 3c). If the medium consists of homogeneous layers of contrasting den-sitv and, or velocity, waves incident at an interface generate reflected waves and transmitted waves;which waves arrive at the receiver is determined by the angle of incidence and the number of times

the waves are reflected (Fig. 3d).A P-wave or a St'-wave that is incident obliquely at a planar boundary between two media in

welded contact generates two reflected waves (P, Sf) and two transmitted waves (P, SF). The

angle at which each reflected or transmitted wave leaves the interface is determined by a general-ized form of Snell's law. Figure 4 shows a P-wave ray in the x-z plane arriving at a planar bound-ary with an angle of incidence, 0Pinc . By Snell's law,

sin Oin sin 0 sin 0Srff Siln OPrfr sin OSrfra- l - - a2 - 02 constant (2)

where inc, rl and rir denote incident, reflected and refracted (transmitted) waves, respectively. - -

The constant is termed the ray parameter p. If a SF-wave in the x-z plane is incident at an angle -"'

0 sic, the governing equation is

lpsin 0si sin 0S sin 0, sill 0S sin 0 prfr

- -sn -- rf - -I'f 02f - f a constant -(3)

0 01 a1 02 Q2

A S//-wave in the .v-i plane incident at the boundary does not generate P-waves or S V-waves be- - .cause its motion is parallel to the boundary. Equations similar to eq 1 mauy be written for thecompressional wave potential and the shear wave potential in each medium. The potentials aresubstituted in equations stating the continuity of stress and of displacement across the boundary:the set ot equations is then solved for the coefficients of reflection and of transmission (refrac-

tion). [ach coefficient is a ratio of amplitudes (of two potential equations) and depends uponthe angle of incidence of the waves and on the elastic properties and densities of the media. Co-efficients may be obtained also in terms of the amplitudes of displacement or energy. Graphs of

4

S R R Figure 3. i1are types i hountded and un-

o b hounded inedia. S denotes source, R de-notes receiver.

a. Direct wale in itninite,h(oillogetllcolis tiledilmlt.

b. Surface-rellected wave.s R c. Ref racted wares.

d. Mulrip/v reflected waves

C d i a layered nedium. -

-. 5.

I€ Figure 4. Rai, for each reflected (P, S) plane3 i€wave and each transmitted (P, S) plane wave

:, 2> a generated hy a plane P-ware incident at an

P 2 2 3, angle 0 tpinc at a planar boundary. The P-2 and S-wave velocities are a, and 01 in Ied-

iumn 1 and a2 anld 02 hi medium 2.

tr.nhsmission and reflection coefficients as a function of angle of incidence (for various density and

,eIocit, contrasts) are given in Grant and West (1965, pp. 57-59) and Telford et al. (1978, p. 254). -

As the angle of incidence of the P-wave increases from 0,in = 0, the angle of refraction of the

transmitted P-wave also increases (eq 2). When 0pine = sin -1 (al/a 2 ), 0 Prfr = 90 0 and the trans-tuitted I-wave propagates in niedium 2 at grazing incidence 0Pine is termed the critical angle

(0P,,rit) for refraction of the P-wave and the refracted wave is termed the head wave. The analysisof head waves is based upon the theory of curved wave fronts at a planar interface (or plane wavesincident at curved interfaces). The reflected P-wave energy is maximum at 0Pin

= because0[in, Pcrit -

thus is tle condition for total internal reflection.

When 0Pinc 0crit' Snell's law would require (a2 /al )sin0pi be greater than 1. In this case,

,rrt is taken to be complex, 01,rf r = ir!2 - i6 and 6 = cosh-i [(a22/a,)sinO C (Grant and West1)65. p. 60). When O''rfr is substituted in a potential equation for the transmitted P-wave, the

equation that results has the form .

1'rtr a l I'eI" cxp ( . ' V ""1

s here S .

= a(01 a2 2,'.(i -1

is an imaginary number. Therefore. the transmitted P-wave as expressed by (Pl, rfr is not an ordi-

nar, /.wave. hut rather is an inhomogeneous wave characterized by an exponential decrease in

amplitude w ith vertical distance from the interface. In terms of the ray parameter, the transmit- - -7-

ted P-wave Is anI inhotlogeneous wv'e whet p > I /a 2 . If 02 < aI, a P-wave incident at an angle

'i ri c i'crit gencrates a transiti it ted SI -wave at a real angle of refraction and so tile S '-wave

propagates a ,av fr,ii tile hmdat . If P2 >al , there is a critical angle for refraction of the S 1-

e n 'ell Ot - crit - Sn (al 02 ), tile S i-wave generated by the incident P-wave is aieid wave. If A c ~ " "rit. the angle ,)f refraction of the Sl'-wave is complex and the trans- '

tlltted ,I -w, ave is an inirttMIigeneous wave. Both P and SI' transmitted motion are trapped at the

intertace.

-x1+ Ar'flnodes

z

/+

Sin 8,z (n-Hi)-

+ \ /+ + + + .

+ + + /+

+ + + + -,,,

+ +

__ -

+ + +, X+

*+ /*. *, \'/ ++

/"

+ + + +-- + + + _e.

igure 11. Plane war'e representation of/normal modes of'0the Jirst three orders. The houndarl- conditions are a pres-sure-re/eased surI~ce and a pre ssure-reinfo rem en t (rigid)h(-tom (adapted from Uriek 19 79).

decay with range. If the imaginary part of the complex wave number is much smaller than the

real part, then the real part is used in determining phase velocity (C = V/silly" w/k,,) and group

velocity W(U dw/dk,), while the imaginary part is a damping coefficient (Brekhiovskikh 1960, p.

405). Retlection coefficients that are dependent on 'y are used when the waveguide boundaries arenot perfect reflectors.

The approach used above to determine the condition for constructive interference of plane

waves in a homogeneous waveguide can be applied in the case of a layered waveguide (velocity de-

pendent on depth within waveguide). This requires that the effect of partial reflection at the lay-

er interf ,aces on the phase of each plane wave and onl the energy content of the plane waves be in-corporated in the solution.

Normal modes that have S1! polarization are Love waves, which are specified by their mode

niumber (it = 0, 1. 2. -j. The fundamental normal mode with S V polarization corresponds to the

dispersive Rayleigh wave and is designed as the Rayleigh miode: the higher order modes are desig-

nated as thle first, second, etc., S V mode (Grant and West 1965).There are main computer programs available to obtain normal mode solutions to underwater

wave propagation problems, some of which are useful in solid earth studies. The normal mode

solution can he obtained by solving thle llelmiholt7 equation: this is a timec-independent wave equa-tion that results when the pressure field is assumed to be ofconstarnt angular frequency, P = S(r,z)

exp -iwta. It' the spatial iactor has the form S p(klr)(r, z and thie asymptotic approxinia- - ,

tion of the Ielankel function (valid for A r I is Substituted, then the llelguio7 equation thatresults in clindrical coordinates) is

~2, _ 2 2' 2+ + 2iA o + (k - , =

penen )r dethwihi waeud) hsrqie htteefc f ata elcina h a-B"-'

er ntefacs o th phse f ech lae wve nd n te eerg coten ofth plne ave bein-i'. "1 () "]

corpratd i thesoltio. ':i': S

(nodes), the lines of positive signs are wave fronts of positive wave amplitude, and the lines ofnegative signs are wave fronts of negative amplitude. Each wave front inpinges oil the waveguide

boundaries at all angle -y. In order for superimposed plane waves to interfere constructively, tile

phase difference between tile waves must be an integral multiple of 2r (2nir, n = 0, 1, 2, ... This

requirement leads to a relationship between thickness of the waveguide, angle of incidence of the

plane waves, and frequency. Only waves of certain frequencies will interfere constructively andgive rise to a standing wave pattern in the z-direction; frequency is uniquely determined when

mode number (In) and -y are specified. The apparent velocity (velocity in the x-direction) of the

plane waves is C = V/sinY", this relationship is used to rewrite the equation for constructive inter-ference in terms of C instead of 7. The result is a dispersion equation (phase velocity-frequency

relation) for each normal mode. 0.

Since a critical angle of incidence exists for the retention of wave energy in the waveguide by

total internal reflection, there is a cutoff frequency for each normal mode. At frequencies below

the cutoff frequency, the normal modes are attenuated as they prop ,eate even in a lossless medium

due to the transmission of wave energy across the waveguide boundary (leaky modes). The energylost from the waveguide as transmitted waves may be returned by reflections at depth and so, over

short ranges, may contribute to the wave field at the receiver.

This treatment of normal modes requires that the wave energy has propagated sufficiently farfrom the source for the representation of normal modes as interfering plane waves to be valid. The

reflection coefficient for spherical waves is the sum of the plane-wave reflection coefficient and a

correction term that goes to zero at very high frequencies. The correction term is negligible when

source and receiver are located at distances from the waveguide boundary that are large relative to 0wavelength (Brekhovskikh 1980, p. 247; Clay and Medwin 1977, p. 64). In a mathematical deriva-

tion of the normal mode solution, the asymptotic representation of the ttankel function is substi-

tuted for the function itself, which is valid for distances from the source that are large relative to

wavelength.

The normal mode solution for the relatively simple case of a fluid layer over a rigid half-space -

(Fig. 10) is given here as an example of the solution format. (The case of a solid layer over a rigid

bottom is more complex because two types of waves are present [P, S] and so two potentials are

required to express the normal mode solution. If the half-space is nonrigid, a solution must be

found for that region as well, e.g. that of Ewing et al. 1957, p. 138.) The potential 0 for horizon-tally propagating waves in the layer is

0= 1 Bn [R 1 (-Yn) exp(-ikz cos-yn ) + exp(ikz cosy7)1 exp(ikx sin-,n ) (13)

(Brekhovskikh 1960, p. 407), where n designates a particular normal mode, R, is the reflection co-

efficient for waves incident from above at the lower boundary of the waveguide, and the coefficient

B, which is a function of the position and strength of the source, determines the degree of excita-

tion of each normal mode. The z-dependence of the amplitude of each normal mode is determined

by the expression in the brackets of eq 13. The potential 0 is the sum of potentials for the upgoing

(0-) and downgoing (0+) plane waves where 0- = A exp[i (-kzz + kx ), 0+ = B exp[i (k,: + kxx) ] ,

k, = k sin-y, and k, = k cos-y. An equivalent equation could be written in terms of the coefficient

.1 and the reflection coefficient for waves incident from below at the upper boundary of the wave-

guide, R 2. The potential 0 (eq 13) satisfies the equation RIR 2 exp (2iklt) = I. Examples of

plane waves at three angles of incidence and the normal modes that result from constructive inter-

ference are given in Figure I I. For a given frequency, a higher order mode corresponds to a steep-

er ray path.

The nornial mode solution to the wave equation can be generalized to include attenuation due

to propagation in an imperfectly elastic (lossy) medium and imperfect reflections at the waveguide

boundaries this is done by making tile wave number complex and so introducing an exponential

18

• S.

. . . . . . . . . . . . . . . .

Consider a homogeneous elastic layer of thickness /f over a homogeneous half-space, each withshear wave velocity (0) and density (p) as shown in Figure 8. The dispersion equation for Lovewaves is

tan .2 T -) (c/ 1) (12) .

its solution is graphed in Figure 9. The cutoff frequency is zero for the fundamental mode (n = 0)and

for the higher modes (n > 0) (Aki and Richards 1980, p. 264). For a given mode, the phase veloc-ity is equal to 02 at the cutoff frequency and is equal to 01 in the linit of w- (asymptotic solu-tion to eq 12). This is physically reasonable in that high frequency waves (short wavelengths) areconfined to the vicinity of the layer surface whereas low frequency waves (long wavelengths) pene-trate the deeper, high velocity medium and are little affected by a relatively thin surface layer. Adispersion relation for Rayleigh waves is more complicated. Pilant (1979), Takeuchi and Saito(1972), and Aki and Richards (1980) treat Rayleigh wave propagation in a layered medium. Equa-tions I Ia, b indicate that Rayleigh waves are retrograde elliptical at the surface and prograde ellipti-cal below the nodal plane. Results of computer calculations of surface wave dispersion indicatethat higher order modes are prograde elliptical at the surface.

The characterization of Rayleigh waves and Love waves in terms of modes indicates a relation-ship between surface waves and normal modes: the relationship will become apparent in the discus-sion of normal mode theory in the next section. Here, it should be noted that certain conditionsmust be satisfied if the approach of modeling surface waves as plane waves that are multiply re-flected in a layer is to be valid. The use of plane waves requires that the waves have propagatedsufficiently far from the source to be well represented as planar rather than spherical or cylindrical.This condition is expressed in terms of the layer thickness: the distance from the source must belarge in terms of the layer thickness (Grant and West 1965, p. 77).

SECTION 5. NORMAL MODES

A discussion of normal modes is given in Ewing et al. (1957), Grant and West (1965), Officer(1974), and Brekhovskikh (1980). The propagation of normal modes can be represented as arisingfrom the interference of plane waves that are multiply reflected at the boundaries of the waveguide.

One set of reflected waves in an elastic waveguide with plane, parallel boundaries overlying a rigidhalf-space is shown in Figure 10. The solid lines are wave fronts of zero amplitude of the waves

..- )A+ .. _ .7.

2 C2

f1igurc 10. Wave ronts impinging m h(,tndarics ,t waie-uti at angle -Y(adapted .,jr, (:rick /9S.?),

17

(u) group velocity(c) phase velocity

a2 - - - - - - - - - - - - - -

U c

-a

Frequency

Figure 7. Dispersion curves for three modes (1, 2, 3/. Thecutofffrequencies are we, Wc2, and cwc respectively.

The curves are drawn for the situation of increasing seismicvelocity with depth (adapted from Officer 1974).

P, HC =B, Left hond of eQ 12 C =12

P2 )3 2 \ "

1 ' 2 > 9 Rootsof

Figure 8. Geologic model for which the dis- H if3 -4,2/c2

persion equation for Love waves (eq 12) is __,

calculated.S 2 :3 4

Surface waves propagating in a layer canbe viewed as arising from the interferenceof plane waves that are multiply reflected Figure 9. Graphic solution of eq 12 for the

at the boundaries of the layer. The condi- dispersion of Love waves in a single layer

tion for constructive interference of the over a half-space. Each side of eq 12 is plot-

plane waves leads to the development of a ted as a function ofH (13i2/C2 J "/10

dispersion equation that expresses the fre- (adapted from Aki and Richards 1980).

quency-velocity relation of solutions to thewave equation in terms of the thickness of the layer, the velocities and densities of the layer and

substratum, and the angle of incidence of the plane waves. The dispersion equation is multi-

valued due to a tangent function; one dispersion curve is obtained for each mode (value ofn,where n = 0, 1,2 ... , each corresponding to a range of the tangent function) specified. There is

a cutoff frequency wnfor all but the fundamental (n = 0) Love mode and for all modes in a lay-

ered acoustic medium. At frequencies equal to or greater than the cutoff frequency, surface waves

propagate horizontally with essentially no attenuation in a lossless medium but do decay exponen-

tially with depth; at frequencies below the cutoff frequency, the wave attenuates as it propagates

even in a lossless medium due to the radiation of energy across the layer boundary (leaky modes) 9as well as decaying exponentially with depth. The cutoff frequency increases with mode number

so there is a finite number of modes possible at a given frequency. The phase velocity (c) is the - .

velocity at which a point of a particular wave is propagating. The group velocity (U) is the veloc-

ity at which the energy associated with a particular frequency group travels. Group and phase

velocity are related by U = c - X(dc/dX). A generalized example of dispersion curves for phase

velocity and group velocity is given in Figure 7.

16

0 ?

and

= B expiw [t - (xc) + bzJ. (9)

The amplitude coefficient B must be directed in the.v-direction, so eq 9 is rewritten as

y B expio,, [t - (xlc) + bz]. (10)

Substitution of eq 8 and 10 in the wave equation (a2/3 2 = V2 (a 2/aX 2 ), V = a or 0) leads too2 =a

I1/a2 - I/c 2 andb 2

= 1/02 - I/c 2 . Ifc>a>0, aandb are real and 0and P represent two waves•Ypropagating in oblique directions that do not decay with depth. In order to obtain a representa-tion for waves that are restricted to the vicinity of the surface, it must be that c < a and c <3so that a and b are imaginary. By choosing the imaginary parts of a and b to be positive (a -il /c - I/a 2 , b = i/1/cr - 1/02 ), the waves will decay exponentially with depth as they propagate.The system of equations that results from application of the boundary conditions a, = Oxi = 0is solved for c; for a Poisson's solid (medium with Poisson's ratio of /4), c =C 0.91940. The dis-placement vector u for Rayleigh waves has scalar components:

c-M expiw (t ) (e 0 • - 0.5773e - 3933 t2z/c) ('"-a)C

u - iM expiw (t -x) (-0.8475e - °.' 4 75wz/c + 1.4679e- ° 39 33w~z/c) . (I lb)

C

These equations show that a surface wave of low frequency decays more slowly than one of highfrequency. At the surface, ufju x = 1.468 and at a depth of 0.193X, u, = 0 (Kolsky 1953, p. 20-21). A specific Poisson's ratio has been substituted in the general equations in order to make theequations given here more tractable; alternately, Grant and West (1965, Fig. 3-10) show Uz/Uxand the depth to the nodal layer as a function of Poisson's ratio. .=

In a similar manner, the displacements due to a Love wave propagating in an elastic layer overa half-space are uY = A expiw (t - (x/c) + b I z) + B expiw (t - (x/c) - b I z) in the layer andu =

C expiw (t - (x/c) + b2 z) in the half-space, with b, = V/l/0p2 l/c2 and b2 = /C 2 1/02. The -- .

boundary conditions are continuity of stress and displacement at the layer/half-space interface andthe vanishing of the stress at the free surface. The equation that results upon elimination of A, B,and C is the dispersion equation, expressing the dependence of velocity of propagation on frequency.

The treatment of surface waves can be generalized to the case of several layers over a half-space. - _In each layer there are upward and downward propagating (and decaying) waves that interfere; - 'only downward propagating (and decaying) waves exist in the half space. Long wavelength wavespenetrate the medium more deeply and so have a phase velocity that is an average of the mediumproperties to a greater depth. Boundary conditions on motion within the vertically inhomogene.ous medium are continuity of stress and displacement across the layer interfaces. In order to con-struct a dispersion equation for the multi-layered medium, the motion at a subsurface layer mustbe related to motion at the surface; the equations expressing this relation can be solved by numer-ical integration (e.g. Runge-Kutta method) or by matrix techniques (e.g. Thomson-Haskell method)(Aki and Richards 1980, p. 270). Phase velocity-frequency curves are constructed from the dis-persion relation; a comparison of the curves and field data indicates how accurately the modelused to construct the curves represents the velocity profile and near-surface structure of the earth.

Ftermann (1978) gives annotated computer programs for surface wave seismology. A combina- .-

tion of programs supplied will 1 ) solve for phase and group velocities as a function of period forLove and Rayleigh waves in a multi-layered medium, 2) calculate the Rayleigh wave and Love wavedisplacements in the layers, and 3) generate synthetic seismograms. . .

15 %

. . . . . ... . .

.~~~~ . .

.-.- .--

the computational advantages of evaluating the frequency integral first and then integrating overreal values of the ray parameter (or slowness). Using Chapman's approach, Dey-Sarkar and Chap-man (1978) obtain the following displacement equation valid for (c.rA >> I):

ii(t) * hm[A(t) * I p/C(p) () ]u(t. , - ir(2rs sinA)'/ t=O 4hC2s (psqsPrqr), q, o I

where r = radiusr, s = receiver radius and the source radius, respectively

rn(t) = the source termC(p) = the product of all reflection and transmission coefficients defining the rayA(t) a time series

q --a function of ray parameter p and radiusO(p) = prA+r(p)(p) =the intercept function obtained from a graph of travel time vs range

A the angular separation between source and receiver measured from the center ofthe earth.

The summation is over all real solutions of t = O(p).

Choy et al. (1980) compare synthetic seismograms generated by the reflectivity method andby full wave theory. An advantage of the full wave theory method is that no truncation phase ispresent in the seismograms, while integration over a limited range of angles of incidence causes atruncation phase to be present in the reflectivity seismograms; a disadvantage is that the raysmust be specified in order to be included in the full wave theory calculations (unlike the reflec-tivity method in which all internal multiples and conversions are automatically included).

SECTION 4. SURFACE WAVES

Surface waves exist in the near-surface region of a medium having a free surface. They decayexponentially with depth in the medium but attenuate laterally with a 1/(ranger/ dependence.The types of surface waves are distinguished by differences in particle motion and in conditionsfor occurrence. Love waves are SH-waves that propagate in a layered medium. Rayleigh wavesare coupled P-SV waves that propagate in a layered medium and at the surface of a half-space.In the case of a single layer over a half-space, Love waves exist only when the shear velocity in the

Alayer is less than that of the half-space. Love waves can exist in a multi-layered medium in whichvelocity generally increases with depth, i.e. even if local low velocity layers are present (Pilant1979, p. 161; Ewing et al. 1957, p. 224). Rayleigh waves propagate without dispersion at the sur-

face of a half-space and, like Love waves, with dispersion in a layered medium. However, in thecase of a layer (shear velocity 31 ) of thickness H over a half-space, the Rayleigh waves are not dis-persed but rather propagate with a velocity of C, -0.920, if X << H for all wavelengths (Pressand Ewing 1950).

The nature of Rayleigh waves is evident in the derivation of the displacement equations forwaves propagating along the surface of an elastic half-space. The displacement vector for a propa-

gating wave is u. A property of vectors is that any vector can be written as the sum of a gradient

and a curl; therefore, u = v + w where v = grad i and w = curl i. The potentials 0 and 4' can beexpressed in general terms as*

0 = A exp iw [t -(x/c) + azj (8)

*F.A. Okal, Yale Uni;crsity, personal communication, 1979.

14

.2S

F.~~V T .

are considered: layers i > m compose the reflecting zone for which the complete complex reflec-tivity is R, (w, -); v \lk 2 -kl sin 2 -/; and range A is the angular separation between sourceand receiver positions measured fromm the center of the earth (horizontal range at the surface isx = 0.017 Are where re is the radius of the earth.

In evaluating displacement with this method, a Bessel function typically is replaced by its .717asymptotic approximation for large arguments; this is valid for large source-receiver distances orlarge horizontal wave numbers, i.e. high frequencies. The integration is usually carried out overa limited range (due to computational costs) of real ray parameters. The use of real ray param-eters restricts the method to body waves; the use of a limited range of ray parameters introduces -

a truncation phase in the seismogram. The range of ray parameters corresponds to an equally re-stricted range of angles of incidence; in the case of a deep reflecting-refracting zone, horizontallytraveling energy is left out of the computation. If the top of the reflecting-refracting zone is atthe surface, horizontally traveling energy contained in reflected and refracted waves as well assurface wave energy should be included in the computation.

Burdick and Orcutt (1979), in a comparison of the reflectivity and generalized ray methods,cite the approximations customarily made and reference papers in which such approximationsare not made in obtaining a solution. Sato and Hirata (1980) evaluate results obtained with thesemethods and with a third hybrid method having characteristics of both.

An approach similar to the reflectivity method is given by Kennett (1979, 1983) for planewaves from a surface source that are multiply reflected during propagation in a multi-layered

* medium; effects of excluding converted shear waves and free surface reflections and of includingattenuation are shown.

The final wave theory considered is full wave theory. This is actually a term that applies toseveral theories. The only distinction among the theories given here is that one approach to a fullwave theory of body wave propagation results in the calculation of the WKBJ seismogram.

WKBJ theory refers to a method of finding approximate solutions to a second-order differ-ential equation. When the method is applied to the wave equation, it results eventually in theWKBJ seismogram. Several approximations are made in the process of obtaining the WKBJ solu-

* tion, one of which is to retain only the first two terms of a binomial expansion. As a consequenceof the approximations, the solution is valid for frequencies that are large in relation to velocity

* - gradients or equivalently, for a nearly homogeneous medium. Grant and West (1965) show theequivalence between this asymptotic solution to the wave equation and the approximate solution

* obtained by means of the eikonal equation, both of which involve the propagation of wave frontsalong paths that are determined by Snell's law. The WKBJ method cannot handle the situationof a turning point, i.e. when the ray has reached its maximum penetration depth and is directedhorizontally prior to turning toward the surface, due to a singularity that arises from the use of -

the truncated binomial expansion. A different method, such as an Airy function, is used in thevicinity of the turning point. The solutions obtained are oscillatory above the turning point andexponentially decaying below the turning point.

The full wave theory (Aki and Richards 1980, Choy et al. 1980) is applicable to body wavepropagation in situations involving radial inhomogeneity, smoothly varying density and velocity,ray paths with turning points (including those near grazing incidence at an interface), criticallyincident rays, caustics, wave interference due to multipathing, and non-ray theoretical effectssuch as diffraction. As a result of the first two characteristics, a realistic earth model can be used,rather than a representation of planar layers of constant velocity and density. The great versatilityof the full wave theory results from the incorporation of the frequency dependence of the waves.The equation for radial displacement is integrated in the real frequency domain and in the rayK. parameter domain; depending on the path of integration in the ray parameter domain, the dis-placement equation reduces to the solution obtained from ray theory if the reflection and refrac-tion coefficients are assumed to be independent of frequency. Chapman (1978) has investigated

13

- -. .. .. .*

with amplitudes lower than a preselected percentage of the maximum amplitude of the first ar-

rivals can be excluded from the seismogram. Krebes and Hron (1981) show that there is good

agreement between synthetic seismograms obtained using asymptotic ray theory and those ob-

tained using the Thomson-Haskell matrix method in the case of anelastic media. The method can

be used with laterally varying structures (McMechan and Mooney 1980) provided that the rays can

be traced through the structure.

Quantized ray theory (Wiggins and Madrid 1974) and subsequently disk ray theory (Wiggins

1976) were developed as modifications of asymptotic ray theory to obtain more accurate body

wave amplitudes. Examples of synthetic seismograms obtained with the disk ray method for

models of laterally heterogeneous elastic and anelastic media are given by Braile et al. (1983). ,.A

The third wave theory considered is the reflectivity method (Fuchs 1968; Fuchs and Muller

1971; Aki and Richards 1980, p. 393-404). It is used to study the reflection and refraction (in-

cluding critical refraction) of spherical waves incident on and propagating within a zone of layers

with planar interfaces. The method is customarily applied to body wave problems but can in-

corporate surface waves. It involves the determination of the progressive partitioning of wave . -.energy due to one or multiple reflections of each wave considered together with phase changes

(delays) due to the propagation of the wave through each layer. The layered zone in which re-flections and refractions of interest occur can be located at the surface (Fertig and Muller 1978)

or buried; in the latter case, waves from a source position may be propagated to the reflecting-

refracting zone and returned from the zone to the receiver position by considering only the oc-

currence of refraction (Fuchs and Muller 1971, Aki and Richards 1980)or reflection and conver-

sion of waves propagating in the surface region (Kohketsu 1981). An equation for displacement

at the receiver position is obtained after two integrations, the first in the domain of the wave

number (k), ray parameter (p = k/co) or angle of incidence and the second in the frequency (CO)domain. The reflectivity term in the displacement equation is the exact sum of the infinite series

of reflections for all possible multiply reflected waves (including converted waves) within the re-

flecting-refracting zone (Aki and Richards 1980, p. 402). Accordingly, the solution obtained

with the reflectivity method is equivalent to the net displacement due to the arrival of an infinite

number of generalized rays.

The compressional potential obtained from the reflectivity theory for the case of a point source

at the surface of a plane layered earth is

) F() k sin'C' c exp(-i(Ak& sin- 4 -

ivl \/21r~k,, siny 4-

ST(w, -)" R p (,, y) exp(-ir) dy

(Burdick and Orcutt 1979) where

F(w) = the Fourier-transformed source time function

t = angle of incidence in the rth layer: = w/a . -'

The indexi refers to the ith layer; layers/= 1, 2 .... m compose the refracting zone in which only

the transmission losses T(w, y) and the phase shifts r(w, -t) of a plane wave at constant y and w

12

. :

in order to reduce computational time; here r is the travel time of the ray with ray parameter p

and the potential equation is

1 1 Rc(r, Z, t0 = H (t - '""' •I.

where R = sIr2 + z2 . Similar equations for transmitted waves and reflected waves in a layered

medium involve the potential transmission coefficient and the potential reflection coefficient,

respectively, and an R term that is dependent on the location of the source relative to the layer

boundaries.The displacement field at the receiver position is determined by summing in the time domain

the contributions of rays that travel from the source to that point. Each reflection or transmission

of a ray at a layer interface is taken into account as it proceeds from source to receiver. Computa- " -

tional cost may impose a limit on the number of rays included in the construction of the seismo-

gram. A first attempt is to use only the rays that are reflected once between source and receiver;

the number of such rays depends on the number of layers used to model the velocity profile of the

medium. As multiply reflected rays of increasing order are included in the computations, the syn-

thetic seismogram changes until eventually inclusion of additional rays does not further change the

character of the seismogram. For a medium composed of thin layers, the number of rays that must

be included for accuracy is high; in this case, truncation of the infinite series of multiple reflections

can introduce serious errors. The multiples that must be included are specific to the problem and

depend on the depth and thickness of the reflected layers, the velocity and density contrasts at

interfaces, and the source-receiver separation. The validity of the synthetic seismogram genera-

ted depends upon the incorporation of all necessary rays. The solution is more difficult to obtain

if attenuation is incorporated in the problem.

The second wave theory considered is asymptotic ray theory (Hron and Kanasewich 1971,

Pilant 1979). This approach represents the displacement due to propagation of a plane wave in a

layered medium as the sum of an infinite series of products involving a spatially dependent ampli-

tude and two frequency terms, one an exponential and one a power series in inverse frequency,

0=0 e iw (t-T)( x , y , z , t ) n (X , Y , Z)

n=0 (j~)f

here T is a phase function that depends upon the ray path. The result is an asymptotic solution

to the wave equation that has characteristics of geometrical ray theory. In the limit of high fre. ..- '

quencies, a useful asymptotic solution is obtained with only one or two terms retained. If only

the first term (n=0) of the series is retained, the displacement equation is the same as the solution

to the eikonal equation (eq 4) and the results obtained are necessarily identical with those due to

geometrical ray theory for wave propagation in a plane-layered medium. Higher-order, frequency-

dependent terms are used in problems involving diffraction phenomena, curved interfaces, and

inhomogeneous media; the n=l term is necessary in the case of head waves.

Hron et al. (1974) give examples of synthetic seismograms generated by the asymptotic ray

method for the case of a compressional plane wave impinging on a plane-layered medium and asurface source. Each ray is specified by the number of linear segments composing it and the num-

ber of conversions that occur. An advantage of this method is that each constitutent ray (or group

of kinematic rays) of the synthetic seismogram can be identified. The amplitude of each ray is

determined using plane wave coefficients of reflection and transmission; the use of frequency- ...-

independent coefficients is justified by the use of zero-order terms (high frequency approxima-_-":. "

tion). Geometrical spreading, neglected here because the source is assumed to be at infinity, is

normally incorporated in the ray amplitude formulation (McMechan and Mooney 1980). Rays

.- •%.

1 1 "- - •

; . - . . -.. + . 7 - -d - -,----- . - , .- r. - - - , , . : .. . .. . *: ; - r . . - r-. - r -- - r :-r -r .

formed by the intersection of rays) with finite amplitudes rather than the infinite amplitudes thatray theory predicts, and the interference of waves that travel different paths but have similar ar-rival times (multipathing). The requirement concerning velocity changes (Xo (6C'/C) << 1) issatisfied if the wave is of sufficiently short wavelength (high frequency approximation) or if abruptvelocity changes in the actual velocity profile are replaced by layered transition zones of constantvelocity in which the thickness of the layers is such that the requirement is not violated, i.e. thick-ness/wavelength > 10 (or even 100). The requirement concerning changes in curvature of the wavefront (X. 6 It'" << I), which again is satisfied in the limit of high frequencies, limits the validity ofthe ray theory approach to plane waves this requirement is a restriction to the far-field region suchthat spherical waves have traveled far enough from the source so as to appear as plane waves. Thefar-field condition requires that the horizontal source-receiver separation be equal to several wave-lengths.

Wave theoryThere are several methods of generating synthetic seismograms that are based upon wave theory.

In each case, the wave theory can he shown to encompass geometrical ray theory. A significant

difference between ra, theory and wave theory, which often also formulates its development interms of rays, is that wave tleory incorporates a dependence on the frequency of the seismic wave

in its solutton.- The first wave theoiui considered is generalized ray theory, also known as exact ray theory, the

method of generali/ed reflection and transmission coefficients (Muller 1970) or the Cagniard-de[loop method (Aki and Richards, IQX 224-253, 3,S6-393). It is applicable to problems involvingspherical waves incident on planar interfaces and can incorporate direct waves, reflected waves,critically refracted %aves and Rayleigh waves in the solution. It requires an integration in the com-plex ray parameter plane and a Laplace transform to obtain solutions in the time domain; the pathof integration is different for each generalized ray and must be found exactly. In Laplace space,the compressional potential due to a point source in a homogeneous medium ist

(lr, z, s (s) . o J P K(spr)e-S77 ZI.dp,

where p = the complex wave parameters = an imaginary parameter analogous to the frequency of a wave (p = y/s, where -t is

analogous to a radial wave number)a = X + 2Pr.C = (l/a' _p2)2G = Green's function

K0 = the modified Bessel function.

"- . No approximations need be made in obtaining the solution. If Bessel's function is replaced bythe first term of its asymptotic expansion, the expression for ¢ in the time domain that results is

-R z, t)(t-R la)

Vt ~~ Vvt \ /t2 R2 I

where 11 is the Heaviside function and * denotes convolutiont. The first-motion approximation

,-r + 2pr)V2 - (2pr)2, which is valid for large ranges (r) but fails at long times (t), is often made

t E.A. Okal, Yale University, personal communication, 1980.

10

*7. - .

i. e. when sinO = 1; this is the penetration depth of the ray which then turns upward. The time re-quired for the ray to travel from depth Zo to depth Zf is given by

Zfif _ to = [ dZ - . '-dZ (6) IC(Z)[I - p2Cv()] (6

Z 2The horizontal (surface) component of the ray path at a depth Zf is

X J p (Z) (zXf - X 0

= [ -p C ( ) " (7) ... .;

•[ z0 '--.' I'-

If the ray reaches Zpthen the integration of eq 6 and 7 must be performed twice, once for thedowngoing ray path and once for the upgoing ray path. The accuracy of all the calculations de- -.-

pends both on the approximations of ray theory stated above and on the velocity function C(Z)used to approximate the actual variation of velocity with depth. Greenhalgh and King (1981) givethe ray equations that are valid for various velocity-depth functions. Also, the equations were de-rived for the case of a flat earth that is homogeneous or is inhomogeneous with depth only. Theapproximation of a flat earth is justified in wave propagation problems involving source-receiverseparations small enough that the curvature of the earth may be neglected. Officer (1974) derives

4 equivalent equations for the case of a spherically stratified earth; for this case, the quantity that is ,constant along the ray is r sin O/C, which is also called the ray parameter, where r is the radial dis-tance from the center of the earth.

Shadow zones exist when the velocity profile is such that rays are directed away from a parti-cular region; this occurs when a trend of gradually increasing velocity with depth is interrupted bya negative velocity gradient (velocity decreases with depth) and then resumed. Similarly, conver-gence of rays occurs when the velocity profile is such that rays cross and become focused in a re-gion; in this case, a trend of gradually increasing velocity with depth is interrupted by a greaterpositive velocity gradient and then resumed. A complete discussion of variation in the locationand energy content of rays arriving at the surface due to different velocity profiles is given inOfficer (1974).

The path that a given ray travels through a medium can be determined by ray tracing; this re-quires specifying the takeoff angle (the orientation at which the ray leaves the source) and thevelocity profile through the medium and then applying Snell's law sequentially. Repeating thisprocess for a range of takeoff angles leads to identification of locations at which wave energy ar-rives at the surface, as well as focusing and defocusing of the energy. The converse problem, de-termining what rays from a known source arrive at a given surface location, is more difficult andinvolves an iterative process beginning with a trial solution. In the shooting method, the initialorientation of the ray is specified and subsequently adjusted until the ray arrives acceptably closeto the receiver location; in the bending method, the source and receiver positions are specified and -

the ray path is adjusted (Aki and Richards 1980, p. 727). Procedures for ray tracing when sourceand receiver positions are known are given by Chander (1977); they are applicable to the situa-tion of dipping planar layers. Because these methods yield frequency-independent results, the in-terference of waves arriving at the same time is found by determining the phase change along eachray path and summing the results.

Limitations on the usefulness of geometrical ray theory result from approximations that re-strict its validity to situations in which changes in velocity or curvature are small over a wave-length, and from its inability to predict the following frequency-dependent effects: the arrivalof body waves in shadow zones due to diffraction, the arrival of waves near a caustic (the envelope

9

where E the amount of energy per unit time per unit solid angle emittedby a point source at the origin of the coordinate system

00 the angle with respect to the z-axis at which the ray leaves the sourcex the x-component of the distance along the ray path.

The decrease in intensity due to geometrical (cylindrical) spreading is l/x; the change in intensitydue to changes in the cross-sectional area of the ray bundle caused by focusing is tan 00/(xaO"0)(Officer 1974, p. 214). The assumption that energy does not cross rays implies that energy isneither diffracted according to wave theory nor scattered by inhomogeneities along the ray. Thepresence of wave energy in shadow zones, locations for which ray theory predicts no surface ar-rivals, is attributed to scatterers and the occurrence of diffraction that direct energy into the .

shadow zone (Urick 1983).A ray in a homogeneous medium is traced using Snell's law, .'-

sin 00 /C = p -

where p = constant for a given ray and is called the ray parameter

0o = the angle the ray makes with the vertical to the surface (takeoff angle)

C = the wave velocity in the medium.

If wave velocity changes along the ray, the angle 0 at which the ray is directed must change inorder for p to remain constant. This relationship is given as

sin 0o sin~n ' :sino 0 - 5Ifl0 p (5)

CO Cn

for an inhomogeneous medium modeled as n layers of constant velocity Cn; on is the angle indi- t.A

cating the orientation of the ray in a layer of velocity Cn .As the thickness of the layers goes tozero, C is a continuous function of depth, C(Z), and eq 5 can be written as

sinO0 _ sin 0C(Zo) W()- p

(Clay and Medwin 1977, p. 83; Telford et al. 1976).There is additional information on the propagation of energy that can be obtained from ray

theory (Officer 1974, Clay and Medwin 1977). The relationship between curvature of the ray,

dOldS, and velocity gradient, dC/dZ, is

dO dCS -p Z

where S is the coordinate distance measured along the ray. The ray is a straight line in a mediumof constant velocity; it is a portion of a circle in a medium with a linear velocity profile, in which

case the ray curves toward the region of minimum velocity. The ray becomes horizontal at a depth

Z if

sin 0 = CZ 0)

8

K-is:. ... .......

Solutions of the eikonal equation have the form W(x, y, z) is constant and represent wave fronts.At a given time t, all points along a particular wavefront W will be in phase. A normal to the wavefront along a particular wave front defines the direction of propagation and is a ray. Officer(1974) shows that

U = A(x, )', z) ei- I W(x,Y,z)/C 0 - tl

is a solution to the wave equation if the following equalities are true:

aW 2 aW2 aW2 o2 1 42A 2A a2Aa a+ + n) =0z T A- ax ay aZ

a2w a2W a2w 2 ++ ±)=x + + ax ay a 0.

if2 a2O 1 a2A a2 A a2A

+ay 2

then W is a solution to the eikonal equation; here, X0 = 2 rCOc is the wavelength of the distur-bance at angular frequency co in the reference medium of velocity Co.This condition is satisfied ,t--in the limit of high frequencies (X-+0). Of practical interest, W is a solution to the eikonal equa-tion and U is a good approximation to a solution of the wave equation if X0

2 (A "IA) << (' 1)2

(prime denotes differentiation with respect to a spatial coordinate). This condition can be re-written to show that a solution to the eikonal equation will satisfactorily approximate a solutionto the wave equation if 1) the fractional change in velocity gradient, 6C', over a wavelength issmall compared with the gradient C/Xo, (X0 (6C'IC) << 1), 2) the change of curvature of thewave front is small over a wavelength, (Xo6 W" << 1), or 3) the fractional change in the space rateof change of amplitude is small over a wavelength (X0 (6A '1A) << 1) (Officer 1974, p. 205).

The direction in which wave energy propagates is parallel to the rays. Because it is assumedthat energy does not cross rays (i.e. the energy contained within a bundle of rays [raytube] at thesource remains within the space defined by those rays) the propagation of energy through a medi-um becomes known by determining the paths of rays through the medium (Fig. 6). The space de.-fined by a given bundle of rays changes due to geometrical spreading and due to changes in veloc-ity with depth that result in focusing or defocusing of the rays. Accordingly, the intensity of thedisturbance varies along the ray paths. The intensity at a point along a ray path between sourceand receiver at the same depth is given by

E tan0oI (x, 00 ) x(ax/aoo)

- X4

sXoZ,) Figure 6. Rays A and B emanating from asource Sat takeof angles of 00 and 0o +dOo,

D respectivelY. The rays are linear in a medium.8 ol constant velocity, V. In time At, each ray

travels a distance D = VAt. The location of

A ray A and the wave enerKv associated withz it is then (X 0 +D sin6o, Zo +D coso).

7

IA

When the angle of incidence of a wave

at an interface is real, the coefficients (trans.mission, reflection) for plane waves at planar "

interfaces are real and independent of fre- Squency. The latter property means that a2 a,there is no change in wave shape at the in-

terface; i.e. the departing transmitted and

reflected waves have the same shape as the

incident wave but probably different ampli- Figure 5. Reflected waves and transmittedtudes. If either the incident wave or the in- waves generated by plane waves incident at

terface is nonplanar, there may be a change the waveguide boundaries. At angles lessin pulse shape (a phase change) at the inter- than critical thee are represented by solid

face (Aki and Richards 1980, p. 155). rays. Reflected waves and head waves gener-

The angle at which plane waves are inci- ated by plane waves that are critically inci-

dent at a boundary is crucial to the long- dent at the waveguide boundaries are repre-

range propagation of wave energy within a sented by dashed rays. Reflected waves gen-

layer or a waveguide (Fig. 5). When the erated by plane waves incident at angles

angle of incidence is less than the relevant greater than critical are represented by dot-

critical angle, wave energy is lost across ted rays; the associated inhomogeneouswaves are represented by wavy lines.

each boundary in the form of transmitted

(P, S) waves. When the waves are incidentat the critical angle, reflected waves and a head wave exist and the wave energy is effectively trapped

by the boundaries of the waveguide. When the angle of incidence is greater than the critical angle,the criterion for total reflection is satisfied but some energy is lost across the boundary from theinhomogeneous waves. The loss of energy from the waveguide is significantly lower when waves

are incident at the waveguide boundaries at angles greater than critical.

The wave energy in a layer or waveguide may be described by a sum of normal modes. Each nor-mal mode represents horizontally propagating waves that have the profile of standing waves in the

-- z-direction; it is a solution to the wave equation and satisfies the boundary conditions of the prob-

-" lem. The stress-depth profile of a normal mode may be determined from an analysis of the con-

structive interference of plane waves that are totally reflected within the waveguide.

SECTION 3. BODY WAVES: RAY THEORY AND WAVE THEORY

Ray theory and wave theory are each the basis of analytical approaches to the study of thepropagation of body waves in the earth. Their similarity is that each yields asymptotic solutions

to the wave equation; their differences lie in the approximations used in their development and,

consequently, in the restrictions on their validity.

Geometric ray theoryThe concept of rays arises from the eikonal equation,

.3 aW2 3W2 aw 2 (4)ax y z C2

where C = velocity.C0 = a constant, reference velocity

* n = the index of refraction (Co IC n).

6

p 7: ,1. "

A -:

where ko is an arbitrary constant used to define the wave number k 2 = 12ko2, n being the index of

refraction. f fis assumed to vary slowly with respect to range, then 1a2 Ifar 2 1 << 2k 0 (a3far)l andthe Helmoltz equation is simplified to the parabolic wave equation (a2f+z 2 )+2iko(af/Or) + (k 2 -

k 0 )f= 0. Computer codes that solve the parabolic equation also are used in studying problems inunderwater acoustics. The solution to the Helmholtz or the parabolic equation for particular bound-ary conditions (sound speed profile, source depth, receiver depth) is presented as a plot of propaga-tion loss as a function of range for a specified frequency. Coppens (1982) discusses the use of theparabolic equation and cites errors in the calculated acoustic field that result from the assumptioninvoked in deriving it. One error is that the phase velocity of a particular normal mode obtainedby solving the Helmholtz equation is different from the equivalent term obtained by solving theparabolic equation, and so the spatial pattern of the phase-coherent combination of the pressureterms will be different for the two solutions. The parabolic equation may be used under the con-ditions that the fractional change in velocity over one wavelength is small, and the angle of propaga-

tion with respect to the horizontal is small (Urick 1979).When an integration over wave number is done to obtain an integral solution to the wave equa-

tion, a contour of integration along the real k-axis encounters no poles due to normal modes if themedium is assumed to be lossy. The Fast Field Program (FFP) uses the Fast Fourier Transform tonumerically integrate the solution to the wave equation for the situation of lossy layers over a half-space and then removes the effect of attenuation due to anelasticity. The result is values of pres-sure, particle displacement, or velocity at range points r. = ro + nAr, n = 0, 1, ..., N-I. Whichmodes are included in the solution (normal, leaky) is determined by the range of wave numberskm = k 0 + mAk, in = 0, 1, ..., N-I selected. Kutschale (1973) notes that use of the FFP is quickerthan use of normal mode theory in obtaining a solution to the wave equation because FFP solvesthe integral solution to the wave equation directly while normal mode theory requires that theroots of the dispersion equation be determined and then the corresponding normal modes besummed. Kutschale uses the FFP method to investigate how several factors (presence of a surfaceice layer, rigidity of bottom sediments, source depth, small changes in sound speed profile in upper400 m of water, variations in ice roughness) affect compressional wave propagation in the ocean.

SECTION 6. FINITE-DIFFERENCE METHOD

The basis of the finite-difference method is that it provides an exact solution to an approxima-tion of a differential equation. In seismology it is the wave equation that is approximated. Where-as an analytic method provides a continuous solution to the wave equation, u = u(x, z, t), a numer-ical method provides a discrete solution, u = u(mAx, nAz,pAt), to an approximation of the waveequation. The domain of the finite-difference solution is shown schematically in Figure 12 as adistribution of points in which th, spacing between adjacent points is Ax parallel to the x-axis,Az parallel to the z-axis, and At parallel to the t-axis. An introduction to the finite-differencemethod and its applications is given in many texts (e.g. Richtmeyer and Morton 1967, Lapidus

and Pinder 1982).Boore (1972) has reviewed the use of the finite-difference method in the study of seismic wave

propagation. He cites the advantages of using this method as being the nature of the computations(a program to use the finite-difference method is easily written, is adapted readily to a variety ofproblems, and uses input data that are easily prepared) and the use of transient signals (because ofwhich one computer run yields information on many frequencies). He also cites advantages in thepresentation of results: displacement at a given position as a function of time (i.e the time seriesrecorded by a seismometer), or displacement at a given time as a function of position (i.e. the total

wave field). The finite-difference method is useful in solving problems for which there is no a:.'ly-tical solution or for which the analytical solution is more costly or less adaptable due to the com-

20

*- - . . - - - - - - -

- -- ---- - - - - - - - - - - - - - - - - - -

m,p.l

t "

AX -

m-I,p m,p m. p

a. Finite-difference scheme for problems of one-dimensionalwave propagation. ".

f £ ! mm,np. p*I"

At .-

M-m,n,p 15p _m11np

Z- m,n-I~m-- m,np-I

b. Finite-difference scheme for problems of two-dimensional Awave propagation.

Figure 12. Grids and templates for finite-difference method.(adapted from Lapidus and Pinder 1982).

plexity of the physical model. Use of the finite-difference method is particularly advantageous Iwith problems that concentrate on the near-field region of sources or that involve wave propaga-

tion through layered media in the case where the thickness of the layers is on the order of theseismic wavelengths. Another discussion of the fmite-difference method as applied to seismologythat includes more recent references is given by Aki and Richards (1980). Examples of the ways

in which the numerical solution to the wave equation may be displayed are given in Figure 13.There are two types of finite-difference s-hemes, both of which are recursive. To solve for the

future displacement at a specified position, the explicit approach uses supplied or previously cal--culated values of displacement at that position and neighboring positions for past and present " -

times. The implicit approach again uses known displacements at the specified position andneighboring positions, but for past, present and future times. In the notation used here, the ex-plicit approach uses displacements at time levels (p-I) and (p) to calculate displ zements at time ,level (p+l), while the implicit approach uses displacements at time levels (p-I), (p), and (p+l) tocalculate displacements at time level (p+l) (See Fig. 12). Emerman et al. (1982) investigated the ."-

use of an implicit finite-difference approximation of the wave equation and found it to be less -"'

accurate than the explicit formulation for the same size increments of space and time. Braile et """-"al. (1983), however, compared four implicit formulations (including that of Emerman et al.) ofthe scalar wave equation with the explicit formulation of Alford et al. (1974); their results in-dicate that one of the implicit formulations provides reasonable accuracy with a grid spacing that

is approximately twice as coarse (4 to 5 grid points per wavelength) as that required by the ex-plicit and other implicit formulations, and thus results in a factor of four savings in computer

storage and a factor of eight savings in computer time.

An understanding of the rather complex finite-difference equations given below can be gainedby following the derivation of finite difference equations in the much simpler one-dimensional

21

L -?.'?--?.? ".' --? ? . .' ..... . . '.i.-.' . . .. : ( .i. .? i -'- ..- ' ° :?..'-- .. '. .7 " -- " ' ..' -. 3 .-.-.- ;-

14

I 3 H

28

2 4 42

56

70

I-84

I I I I 98RI2 /3 2 3

2.--/ AS' 112 ".""

/.12 $.I ,, .

-150 0 150 300 450 600 750Distance (kin)

a. Geometry and media labels for fburmaterials joining along a vertical interface. b. Surface displacements at different instantsIf the two conditions at the bottom are of time for the problem indicated in a. Thesatisfied, an analytic solution to the prob- location of the vertical interface is given bylem of an incident Love wave can be found. the dashed line. The various elastic and geo-

metric parameters, 01 = 3.85, p1 = 3.0, 24.75, P2

= 3.65, 0j3 = 4.23, P3 = 7.44, P34 =

5.53, p4 = 8.07, H = 35.0, were chosen suchthat a large reflected wave would be present.The units of/ Pand p are km/s and g/cm 3.

r.'-84 vo w

I I I I I

450 750

Horizontal Distance (kn)

c. Computer-generated contour plots ofthe spatial distribution of Love-wave displacements atdifferent instants of time for the problem illustrated in a. The results here are derived from the

* same experiment as those in b. The top edge of each plot corresponds to the free surface. The Overtical line denotes the vertical interface. The horizontal interface is shown in plot for T = 112.Vertical variation is distorted due to variable grid spacing with depth. Each plot can be viewed asa contour map of the displacements at each grid point for a certain instant of time, in which thearea between every other contour is indicated by a given symbol.

Figure 13. Application of the finite-difference method (from Boore 19 70).

22

..................

case. The x-axis is divided into segments of length AX that are bounded by grid points. The gridpoints are specified by mAX, where m = 0, 1,2, ..., and displacement at grid points is given byU(mAX) = Um.

Finite-difference approximations to the partial derivatives au/ax = Ux, a2 u/ax 2 = U, are ob-

tained from a series expansion of U(X). Two Taylor series for U(X) about the point (mAX)Xm are

" U(Xm+A IMU(Xm) +AX XIUxxx + +(3 (14)

M.~~~~ + A.i.)+AXU2!3! UXIM+*

(AX) Uxxx + (15)(Xm - X) =U(Xm) - AX x+ x 2 3! Im (1"

Equations 14 and 15 are solved individually for U,Im to obtain

'-"U(Xm + AX) - (Xm) AX (AX) 2 (Ux = AX 2! ---3! UXXIm " (16)

U(Xm) - U(Xm - AX) + AK U.. (V Uxxx + (17)

Xm =X 2! m 3! Im

Two approximations to Ux at Xm are obtained from eq 16 and 17 by omitting terms in which AXis of the order 1 or higher:

U(Xm + AX) - U(Xm) Um+ Um (18)

U(Xm)-(Xm -. y) - Urn Um(19)

VIm AX AX

Equation 18 is termed a forward difference; Ux is determined by the displacements at Xm andthe adjacent, higher index grid point, Xm+i. Equation 19 is termed a backward difference; U,, Imis determined by the displacements at Xm and the adjacent, lower index grid point, Xm." "

A central difference approximation to U, Im is obtained by adding eq 16 and 17 and solving for .

m.- .Um- (20)Im - 2AX

By subtracting eq 17 from eq 16 and solving for U,, 'the following finite-difference approxima- tion to a2 u/ax2 is obtained: -

I Um+1 - 2 Um + UM 1

-

The first term of the series remainder is -((A 2 /12) Uxx e where Xm -A < e < Xm + AX.The equation which Boore (1972) used in his study of Love surface waves and SH body waves

isti+ UPP +UP".-'

UP1 -2uP UP M m + -I2U - np-i + 2 (At)2 m+,n mn m-l,n m,n+- m,n-m,n m~f f,n ~ A + ()

23

S.".'

where uP is notation for displacement at a point (mAx, nAz) at a time pAt and 02 = (/p) 2 .This equation is a finite-difference approximation to the partial differential equation for displace. jment in the y-direction in a homogeneous medium, a2u/at 2 = (32(a2u/ax2+a2u/az2). It is correctto second order in the increments. In the case of an inhomogeneous medium,

-+ 2u' - 1mn = m,n n

S(t) 2 [m+yznUm+i,n (Pm+vz,n +1m- /z,n)Um,n + tm - ,nU m - 1,n

Pmn ) 2 -

+ (At) 2 [1mn+ Um,n+ l - (/m,n+2 + mn-/2) Umn + Mm,n- zUmn-I

Pm,n (Az) 2

(Aki and Richards 1980, p. 781).The wave equation for the situation of in-plane waves (P-SV, Rayleigh) propagating in a homo-

geneous isotropic body is

a 2 au +a(a2 - au

Ft- = - aaaz 3Z 2 x

a2W= 2 aW + (A 2 +02 a2waty -a az r axaz az2

In vector form the wave equation is

a2u a2u aZ2u + au , ,_-- = At-C + BaX a aZ-

where u=() .:w

-( o .2 .

B=( 2 o2 aa-3 0

C= (0 a2)

Aki and Richards (1980) (citing Alterman and Rotenburg 1969) give the following finite-difference

approximation to the wave equation:

u(x, z, t + At) = 2u(x, z, t)- u(x, z, t - At) + A ( ) [u(x + Ax), z, t) - 2u(x, z, t)

+ u(x - A ,z, t) + B [u(x+Ax,z+Az,t) - u(x+Ax,z-Az,t)4A

At 2- u(x-Ax, z+Az, t) + u(x-Ax, z-Az, t)] + C (-) [U(X, Z+Az, t) -

24

. ..

- 2u(x, z, t) + u(x, z - Az, t)].

This finite-difference equation is correct to second order in the increments. Aki and Richards alsogive Alterman and Karal's (1968) finite-difference equation for spherical waves propagating froman explosive source in a layered medium, and a finite difference equation for one-dimensionalwave propagation in the case where wide-angle scattering, back-scattering, and reflection are negli-gible: the latter equation is suited to the study of wave propagation in a laterally varying mediumfrom a teleseismic source.

The wave equation (constitutive equation) can be approximated by several different finite dif-

ference equations that may all approach the differential equation in the limit that Ax (and AZ)and At approach zero, but that may have different solutions when the increments are finite. In..

general, the choice of a finite difference equation to approximate a differential equation affectsthe accuracy and stability of the numerical solution. Equations 18, 19 and 20 are all finite-differ-ence approximations to Ux Im. A measure of the error that results from the finite-difference

approximation is given by the first term of the truncated series expansion of U(X):

AX Xm '<e <Xm + AX(eq 16)u,'

-2 'xx e X,,, AX < e<Xm (eq 17)

- U xI, Xm AX < e Xm + AX (eq 20)F6 Ce

Lapidus and Pinder (1982) list finite-difference approximations to derivitives for the cases of one

and two dependent variables, and list the order of error resulting from truncation of the series.Aki and Richards (1980) evaluate the stability (i.e. does the error due to approximating derivitivesgrow as the numerical solution proceeds, or does it remain bounded?) of various finite difference

schemes for wave propagation in one direction, as given by

au 1 arat P(x) ax

where ti = au/atF = t'(x) aulax

E(x) = X(x) + 2p(x), P-waves/-(x)= l(x), S- waves.

They find that a scheme using a central difference for the x-derivitive and a forward difference

for the t-derivitive is unstable, while a scheme using central differences for both derivitives isstable if restrictions on At and Ax are met. The optimum scheme that Aki and Richards consid-er involves staggered grids, with the grid for u being displaced from the grid for r by half a grid

spacing in both x and t; the use of central-difference approximations leads to an error equal to/4 of the error of the latter scheme above (because the error is proportional to the square of thesampling interval) while having the same stability criterion.

A criterion for stability has the form At < Ax/c, where c is the local wave velocity. For a Ssolution to be stable, the time interval between adjacent grid points must be !ess than or equalto the time required for a wave to propagate over the space interval between adjacent grid points.When this criterion is met, u(m+l, n) is uniquely determined by u(m, n-I), u(m,n+l ) and u(m-l,n). The stability condition for SH-wave or Love wave propagation in a uniform medium of nspace dimensions is tAt/Ax < 1/n (Boore 1972). The stability criterion for the two-dimensionin-plane (P-S V and Rayleigh) wave problem described above is At/Ax < I 1/(a + 0j )v (Aki andRichards 1980, p. 782).

25

', " o - --' . -"I -" - • - - ' - -.• .. . . •' . " ° . .o . - - ." -o

. .•

, - - -.• ., ° • ' - ° o o •

Since the finite-difference method results in a solution to the constitutive (wave) equation onlyat net points, it would seem logical to use a net of small grid size so that tie problem is well re-solved spatially and temporally. The cost to obtain the numerical solution, however, is dependentprimarily on the number of calculations that must be performed and on the storage space required,and so is ultimately determined by the grid spacing. A compromise approach is to use a fine gridsize early in the numerical solution, such as near a seismic source, and then to rezone the grid to alarger spacing as the problem progresses, i.e. at net points corresponding to later times and distant " .positions (Cooper 1971 ). A difticulty of the rezoning technique is that intensive (mass independ-ent) quantities such as pressure or velocity that are initially specified for the case of the individual,snall grid zones must now be appropriately determined for the larger, composite zone. In a studyof Love-wave propagation in a plane-layered medium, Boore ( 1972 ) increased the grid spacingwith depth in order to delay the arrival of waves reflected from the boundary of the region ofinterest, this modification, based on the decay with depth of Love wave arrivals, has had mixedsuccess. The maximum storage space required by an explicit finite difference scheme is 2(NX)(NZ), where NX and AZ are the number of grid points in the x and : directions (Boore 1972).Although each time level requires (N.X) (NZ) storage spaces and although displacements at twotime levels (p. p-I ) are needed to calculate the displacements at the next time level (p+l), once

UP 1 is calculated uP- is no longer needed so storage locations for the (p-1 data may be usedfor the (p+l) data.

Application of the finite difference method to determine future displacement requires thateither the displacement at each net point be known at times t = 0 and t = At, or that velocity anddisplacement at time t = 0 be known. In studying wave propagation in an inhomogeneous medium,it is customary to situate the seismic source in an adjacent laterally homogeneous medium so thatdisplacement at t = 0, At (or displacement and particle velocity at t = 0) can be determined froman analytic solution. Displacement is zero throughout the inhomogeneous region until sufficienttime has passed (determined by distance from source and velocity of homogeneous region) for thedisturbance to arrive at the boundary between the two regions; changes in displacement with timeare then determined numerically as the disturbance propagates through the inhomogeneous region.The theoretical displacements at times t = 0 and t = At that are consistent with a given surfacewave shape are synthesized using a Fast Fourier Transform method (Boore 1970, Fuyuki andMatsumoto 1980).

Wave propagation problems studied by seismologists customarily involve physical boundaries

such as a free surface, interfaces of layered media or the perimeter an inhomogeneous region.Alterman and Loewenthal ( 1972) describe a technique for using finite-difference equations forboundary conditions in order to obtain displacements at a free surface or at an interface. In bothsituations the finite difference net is extended past the physical boundary (free surface, interface)so that the boundary no longer terminates the net superimposed on the half-space or layer. Dis-placements at net points on the fictitious line (the extension of the net) are used in the finite dif-ference equations of motion to calculate displacements up to and including the physical boundary.This approach is used also by Kelly et al. (1976) and Fuyuki and Matsumoto (1980).

Additional boundaries are introduced when the finite difference method is used in order to re-strict the size of the domain for which the numerical solution is obtained in accordance with theamount of computer storage available. These artificial boundaries are sources of reflected waves,which will affect the numerical solution adversely only when sufficient time has passed for the re-flected waves to have propagated into the region of interest. Boore (1972) recommends that num-erical experiments using different distances to the imposed boundaries be done in order to definethe space-time region that is free of contamination by waves introduced by the presence of arti-ficial boundaries. The artificial boundaries can be characterized so that wave energy is absorbed(Fuyuki and Matsumoto 1980) or reflections are canceled due to the superposition of solutionsof opposite sign (Smith 1974).

26

*, S

-. - ..-....... .. . .--

Complications with a numerical solution that arise due to the use of a discrete grid are aliasing(false representation of frequency content) of the continuous time series and dispersion of wavespropagating on a lattice. These complications can be avoided by restricting the frequency-depend-

ent information that is extracted from the numerical solution (Boore 1972). The problem ofaliasing can be alleviated by regulation of the source so that little or no power is present at values 2of wave number above the Nyquist wave number, which is inversely proportional to grid spacing. 0The effects of lattice-induced dispersion can be eliminated by ignoring information at wavelengthsfor which there are 10 or fewer net points per wavelength. Alford et al. (1974) showed that, withan explicit second-order difference scheme, there is little dependence of phase velocity and groupvelocity on stability ratio (cAtlAx) for A1v/X < 0.1, where X is wavelength, at the upper half-powerfrequency of the source. 0

Table I lists examples of problems for which synthetic seismograms were generated using the

finite-difference method.

Table 1. Applications of the finite-difference method to seismology.

Problems studied with finite-difference method Wave type/source Reference

Lateral heterogeneity: Love wave Boore (1970, 1972)uniform layer of nonconstantthickness (w/planar free sur-face) covering a homogeneoushalf-space

Pair of layers over quarter-space Love wave Boore (1970, 1972)with vertical interface

Sloping layer Love wave Boore (1970, 1972)

Sloping interface: SH-waves at vertical Boore et al. ( 1971),1) crust/mantle interface incidence Boore (1972) -i2) basin

Semi-infinite medium with sur- Sf1-waves at vertical Boore (1972)face topography incidence

900 wedge (quarter-space) in Line source tangent to Alford et al. (1974)otherwise infinite homo- apex of wedge. In-geneous medium cludes case of dif- 3

fracted waves

Problems of interest in exploration Compressional (P-) wave Kelly et al. (1976)tion seismology: line source locatedhomogeneous layers separated within upper layer

by plane, horizontal interface, Waves evident:includes case of a weathered direct P-waveslayer surface reflected

quarter-space embedded in a P-waveshalf-space head waves

fluid (brine, gas) saturated, P-waves convertedI 00-ft-thick sandstone layer into S-waves byembedded in shale reflection

Also heterogeneous case: elastic Rayleigh wavesconstants specified at each diffracted wavesnet point O

Scattering at a trench Rayleigh wave luyuki and Matsumoto(1980)

Layered half-space: cylindrical Pressire pulse Alterman and Loewen-" "coordinates thai (1972)

Impulsive line source located in P-wavesinterior of quarter-space or ininterior of three-quarter-space

27

Table I (cont'd). Applications of the finite-difference method to seismology.

Problems studied with finite.difference method Wave type/source Reference

Heterogeneous medium: spherical Impulsive point sourcecoordiates

Shearing stress acting on a crackin a plane (deep focus earth-quake)

Shearing stress acting on a crackin a half-plane (dip-slip on avertical fault which breaksfree surface)

Impulsive line source located insideone of two welded quarter-spaces

Dislocation source in a homogeneous S-waves, P-waves/ Braile et al. (1983)half-space Dislocation source

Dislocation source in a laterallyheterogeneous velocity structuremodel; dislocation source is lo-cared on a subducting slab

Layer over half-space Love waves Kelly (1983) .5Thickening channel Conversion to body waves Note: data band-passThinning channel evident filtered to isolateIrregular layer/half-space frequency compon-

interface ents so as to displayChannel with low velocity fill modes

over subsided layer and sub-sided half-space

Channel filled with same materialas subsided layer

Channel %% ith high velocity fillover subsided layer and sub-sided half-space

Layer over two quarter-spacesLayer over angular unconformityPeriodic channels in upper portion

of layer

SECTION 7. FINITE-ELEMENT METHOD

There are a number of similarities between the finite-element and the finite-difference methods.Both involve the division of a continuous, two-dimensional structure into discrete portions definedby nodes (or net points), both yield numerical solutions whose accuracy is dependent on the spac-ing between nodes relative to wavelength, and both assume that the displacements of nodal pointsdefine the complete displacement field for the structure. The distinction between the methods isthat the finite-difference solution is obtained at discrete points by approximating the differentialequations that describe a problem involving an elastic continuum, while the finite-element solu-tion is obtained by approximating the elastic continuum as a composite of structural elementswhose internal displacements are known by interpolation. Information on the finite-element - -

method is found in books on solving partial differential equations (e.g. Lapidus and Pinder 1982) " -

and in examples of the application of the method, e.g. structural mechanics (Clough 1965) andpropagation of seismic waves (Lysmer and Drake 1972, Smith 1975).

The finite element-method requires that a structure be divided into a finite number of planeelements that meet at nodes. The commonly used elements are quadrilaterals and triangles. Thenodes need not be regularly spaced; the configuration of the elements can be such that the mesh -..

closely fits an irregularly shaped structure. As with the finite-difference method, there is a trade- -""

off between accuracy (determined by the number of elements, i.e. element size) and cost (deter-mined by the amount of storage and the number of calculations required).

28

- .o

• -; - , "- "-,. - , " - . . *"- -"._ ." . , -'-" "- . "-", -". . "- - . - -.. " - -. "

Q4.'

-, 21A ""-•--. -I

/8,- .''

fu 06 at ].lateral clement are shown for the cases of a) SH or Love war~es "'i-

antd h) P and SV o)r Ra~vheigh wares (froni lrsmner and Drake 19 72). .

An assumption inherent to the finite-element method is that all external forces and forces be- ""-.

tween elements are transm~itted through nodal points. Each nodal point has one or two force corn- .-

ponents acting on it, depending on whether the wave motion considered is one-dimensional (SH '._.-waves, Love waves) or two dimensional (P and SV waves, Rayleigh waves), and has the same num- 1

ber of displacement components (Fig. 14a, b). Three requirements that must be satisfied simul- "...

taneously are those of equilibrium, compatibility, and force-deflection relationship, i.e. the inter- -....nal element forces acting at each nodal point must equilibrate the externally applied nodal force,['"the deformations of the elements must be such that the elements continue to meet at the nodal . '

points in tlhe loaded condition, and the internal forces and internal displacements within each ,

element must be related as required by its individual geometry and material properties (Clough . .196i). Displacements of points within an element are defined by displacement functions in terms "-''

of the displacements at nodes. A linear element is one for which the internal displacements are .' '-

Linearly interpolated from the nodal displacements. Displacement functions are chosen so that '•-the type of element is unchanged as a disturbance propagates through it (i.e. quadrilateral elements""remain quadrilateral), which satisfies the displacement compatibility conditions on the boundaries '

between elements. The numerical solution is obtained either for the case of a forcing function or -.,.

for the case of specified initial displacements and velocities. ....

lDispla,.ements of nodal points are related to forces through the stiffness matrix [K] . If the . .loading is static,".[-

[ALII 61 el (21)

Ilere 181 is a column vector which represents the displacement field; the size of 16 1is determined .'-'o

by. the number of nodal points (A) and by the degrees of freedom at each nodal point ( 1 }is Of " •-.

size .Vin the case of' Love or Stt waves, 2N in the case of P and SV or Rayleigh waves). The column ..'vector t represents the external forces applied to the structure; it is of the same size as "1 6}

Once [K I is known, eq 21 is solved for the displacements 68} If the loading is dynamic (time r,..._

dependent). an inertia term must be included and ,['"

III lJ "j + IAK 6 = IQ (22a),-,,

(Lysmer and Drake 1972). Ifra damping matrix (C) is included, ,_-%

29 -.-.

.-/:.:

-. ..•. ... ----i•-' _ .8 , - . .. . .. . . . . .. . ..., . . ..

• " ' ' 2".."~8 ". . . . ..0, > < .. . . .. . . . .i I " li

").. . ..- " illl : li 'lllb a~ m -,t m ']l :'- : • -. .. . . . ,' ", t , . ,,8,

- -- '

M 6" + [C] 6 + [K] 5 = (22b)

(Smith 1975). Here [M] is a mass matrix. The prime denotes differentiation with respect to time.Once [M] and [KI (and [C], if included) have been determined, eq 22a, b may be solved for thedisplacements 1 f . When the excitation of the structure is assumed to be harmonic, with angularfrequency w, steady-state problems can be solved by the method of complex response (Lysmerand Drake 1972). Assume that IP is a time-independent, complex force amplitude and u4 isa time-independent, complex displacement amplitude. Let

Q Pf exp(iot) (23)

and

6 u exp(iwt). (24)

Substitution of eq 23 and 24 into 22b results in-s

([K] 2 [M ), = IPf (25a)

(K] +iw[Cl - w 2 [Ml)JuJ = jP, (25b)

each of which is a set of linear equations to be solved for displacement amplitude I u . This

approach is correct for problems involving surface waves or for problems involving body waveswhen the wavelength is greater than the scale size. For body wave problems in which the scalesize is greater than the wavelength, eq 22a, b must be solved in the time domain (Smith 1975).Lysmer and Drake (1972) describe how the matrices [MI and [K] are determined for the solutionin the frequency domain. The procedure involves first determining the mass [M1] and stiffness [K']

matrices for each element and then assembling the element matrices to form [M] and [K]. Because[A'] and IK I are determined for a local coordinate system convenient for the configuration of theelement, they must be transformed into a global coordinate system before being added to give [M]and [K]. Smith (1975) uses the lumped mass matrix which is preferred for problems in the time do-main.

As with any numerical procedure, the accuracy of the solution obtained with the finite-element

method must be evaluated. Smith (1975) determined that for the case of a plane wave propagat- _ing a distance of I kn through linear, rectangular elements, attenuation is negligible and disper-sion is I% for a mesh size of 10 elements per wavelength; attenuation and dispersion are each3% when there are 8 elements per wavelength. These results are dependent on the time-stepping

algorithm and on the length of the time step Smith used. Eight to 12 elements and 6 to 10 ele-ments per wavelength are also cited in the literature. Accuracy is also affected by the interpola-tion used to define displacements within an element. The spacing and configuration of the ele-ments can be varied in accordance wifh the resolution needed in a particular portion of the finite

element model. Contamination of the solution by waves reflected from artificial boundaries(sides and bottom of model) is avoided through the use of nonreflecting boundaries or by termi-

nating the solution before the spurious waves reach the region of interest.Geller et al. (1979) determined the dynamic finite-element solutions for displacements and

velocities observed at the surface due to movement on a fault modeled as a dislocation. Their re-suits are compared with the analytic solution in Figures 15a and 15b. The agreement between the

analytic solution and the numerical solution is worse when the size of the elements is doubled be- -'-

cause the resolution due to high frequency components is lost; there is an insufficient number ofelements per wavelength at the high frequencies when the element size is increased. The agree-ment between the analytic and the numerical solutions is better for displacements than for veloc-

30

W W

005

I0 --- ----- Analytic005 .... __......

-005E __ ___ ____ ___ _ - 0.05

0E-005 j . ..0.00.... . FEM AxIlkm

0.Z; . .. . .. . .......... M~~2 m

04 2

02>' 0.05

0.0

0 0 *:DFEM xz2km-0 05 -........... __ __

0 0 30 -0 0 30Dimne (s) imne si)

a. Displacement and velocity at th surface loain the faosrie. Tn fhe fault.

Fiur 15 Caclae vausoArudmtondet oeetaog a gtlterafalt Reu0 r ie o h nltcslto n o ueia ouin ihdf

fere2 elmn Vies(rGertieta. 1)

04 32

ities because the displacements have a lower predominant frequency and are thus less affected bytie use of a discrete representation.

Examples of problems involving wave propagation to which the finite element is well suited arc

given below.Lysmer and Drake (1972) apply the finite element method to the study of the propagation of -

surface waves in a layered medium and in a region of irregular structure. They define a generalizedLove wave to be any free motion involving harmonic displacements of the form 8 = u(_) exp i(wt -

kx) in an infinite n-layered structure with horizontal interfaces. Here ic te wave number andu(z) is the amplitude function. If the function u(z ) is assumed to vary lii arly within each layer,then a discrete representation of the Love wave is

a I u exp i(wt - kx)

where 6 is a column vector containing the displacements 6j,,/ = 1,2 ... , n of the layer interfaces,lu t is a complex amplitude vector with the components uij = 1, 2 ... , ii, and a is a mode partici-pation factor which indicates the contribution of each mode to the amplitude of the displacement.

The assumption of linear variation of the amplitude function is valid if the thicknesses of the modellayers are small compared to the wavelengths of the shear waves in the layer- this requirement maynecessitate introducing more layers than there are actual structural layers present in the model. Theequation of free harmonic motion is

Q11A2 + [(;j w2 [MI)M] } u 0

where the matrices [A I and [GJ are related to the stiffness of the layered structure. Boundary . .

conditions are given for the case of an irregular zone between two semi-infinite layered zones. Theirregular zone has surface topography and layers with sloping or nonplanar interfaces. Vertical in-terfaces separate the irregular zone from the semi-infinite layered regions. A restriction on thethickness of layers (relative to wavelength) within the semi-infinite zones was given above; an addi-tional restriction is that the interfaces between those layers must coincide with nodal points on thevertical interfaces with the irregular zone. Lysmer and Drake (1972) give the results from two

problems, propagation of Love waves across an idealized alluvial valley and propagation of Rayleighwaves through a portion of the Central Valley of California and the Sierra Nevada.

Smith (1975) gives the results of a finite-element study of three body wave propagation prob- . .lems: the amplification of ground motion by surface topography (mountain), amplification ofground motion due to a semicylindrical alluvial valley, and ground motion due to a deep focusearthquake (case of a subducting slab).

Bolt and Smith (1976) use the finite-element method to compute synthetic seismograms forP-waves vertically incident on a massive sulfide ore body and for SH-waves vertically incident on Sa buried salt dome.

McCowan et al. (1977) use observed ground motions from the San Fernando, California(9 Feb 1971 ), earthquake as the displacements in two-dimensional finite-element model of the

fault. They first consider the static case (eq 21) and solve the inverse problem for the force vector.They then use the same finite-element model for the dynamic case (eq 22b) and compute synthet-ic seismograms. Finite-difference equations are used to relate acceleration and velocity to dis-placement in the dynamic problem.

Melosh and Raefsky (1981) utilize the technique of split nodes in order to advantageouslytreat the presence of faults or cracks in a finite element model. In a standard finite-elementscheme, elements that share a node experience the same displacement at that node; the distinguish-

ing characteristic of a split node is that its displacement varies with the element under considera-

tion. In this way, a split node located on a fault has two displacement values, the first (u') ex-

32

S

. 1-.

perienced by the element on one side of the fault, the second (u-) experienced by the element ontile opposite side of the fault. Melosh and Raefsky state that the use of split nodes does not corn-

promise the accuracy or stability of the computation.

SECTION 8. HYBRID METHODS

The characteristics of a problem may determine that it is advantageous to employ a method of

generating synthetic seismograms that is based upon ray theory rather than mode theory, or viceversa. Table 2 (from Urick 1983, p. 122) conpares normal mode and ray theories. A normal

mode approach handles initial conditions well and is useful at low frequencies (below 500-1000!lz depending on the model used), while a method based upon ray theory handles boundary condi-tions well and is appropriate for programs involving high frequencies and long propagation dis-

tances; although mode theory is valid at all frequencies, its use at high frequencies effectively isrestricted to short propagation distances. Hybrid methods that have features of both ray theory -

and normal mode theory may have lower computational cost yet retain sufficient accuracy.RAYMODE, a computer program written for underwater acoustics, is a hybrid method applicable

to studies of compressional wave propagation at a range of frequencies (low to high) in a mediumcharacterized by depth-dependent velocity and a flat lower boundary at which there may be loss

of energy.

Another hybrid ray-mode formulation developed for studies of wave propagation in wave-guides and ducts (Felsen and Ishihara 1979, Felsen 1981 )has been used to construct syntheticseismograms for the case of S1-waves in a layer over a half-space (Kamel and Felsen 1981). This

method employs a truncated ray series, each ray characterized by its takeoff angle at the source,

and a truncated mode series, each mode characterized by the angle of incidence of its equivalentplane waves at the layer boundaries' together, the two series constitute a complete spectrum of

angles while avoiding problems such as caustics by the appropriate selection of rays to be incorpor- .

ated in the solution. Because the total number of generalized rays and normal modes contained inthis hybrid formulation generally is less than the number of rays or of modes required when either

method is employed independently, the computational cost of the hybrid method is lower. This

hybrid method, developed for the case of high frequencies, includes asymptotic approximations

and results in a solution of the wave equation comprised of generalized rays, normal modes, and aremainder; Felsen (1981) notes that if the asymptotic approximations are not used, the hybridformulation is exact. Synthetic seismograms generated by four methods were compared. The hy-

brid formulation consisted of two generalized rays (direct and singly reflected), two trapped modes,

Table 2. Comparison of normal-mode and ray theories (from Urick 1983).

Normal-mode theory Ray theory

Gives a formally complete solution. Does not handle diffraction problems,e.g. sound in a shadow.

Solution is difficult to interpret. Rays are easily drawn. Sound distribu-tion is easily visualized.

Cannot easily handle real boundary Real boundary conditions are insertedconditions. easily, e.g. a sloping bottom. S

Source function easily inserted. Is independent of the source.Requires a computer program, except Rays can be drawn by hand using Snell's

in limiting cases when analytic an- law. However, a ray-trace computerswers exist, and presents computa- program is normally used.tional difficulties in all but simplestboundary conditions.

Valid at all frequencies, but practically Valid only at high frequencies if (1)is useful for low frequencies (few radius of ray curvature is > X or (2)modes), sound velocity does not change much

in a X , i.e. (AI/ V)/Az << I/\.

33 " - -

. . .-.-- ..

'- and a remainder. The method of generalized rays (Cagniard-de Hoop) incorporated 14 rays. Seismo-grams generated by these two methods were indistinguishable. The seismogram obtained by a solu-tion equivalent to the hybrid formulation without the remainder term agreed with that of the hy-brid method for early observation times but became in error at a time corresponding to the arrival -aof the singly reflected ray; from then on, the discrepancy between t',e two seismograms decreasedwith time. The fourth seismogram, based upon the two trapped modes incorporated in the hybridsolution, was initially it, error but approached agreement with that of the hybrid method withtime.

A second type of hybrid method is one which combines analytical and numerical methods ofsolving the wave equation. Shtivelrnan (1984) uses the finite difference method and a form ofasymptotic ray theory to compute the wave field in media having irregular structures. The finitedifference method is used in the vicinity of a scatterer or complex structure because the numericalmethod generally handles wave propagation in heterogeneous media well, whereas the ray methodis unsuitable when scattering from heterogeneities or wave propagation through thin beds is in-volved. The ray method is used to compute the wave field in homogeneous regions or in regionswhere layers are thick relative to wavelength because in such cases the computational cost andstorage requirements of the finite difference method, together with the possibility of numeric dis-persion causing loss of accuracy, cause it to be the less suitable method. The transition betweenmethods occurs at an arbitrary contour along which the wave field of the first method employedis used to define source terms for the second method. Shtivelman gives examples of the applica-tion of this hybrid method to the case of SH plane waves impinging on an interface that is displacedby a right angle step, an inclined step, a right angle projection, a right angle recess, or a thrust fault.

SECTION 9. CONCLUSION

Several methods of generating synthetic seismograms have been reviewed. The selection of amethod for use is based upon considerations of the physical aspects of a problem--complexity ofthe geologic model, wave types to be included in the seismogram-and of computational cost. Thetype of problem for which various methods are appropriate is indicated by Table 3, in which speci-

" "fic computer codes are evaluated by Braile (1982), and by Table 4.

* Table 3. Summary of seismic interpretation techniques (from Braile 1982).

Method Characteristics Limitations

Travel-time Analysis

Geometrical Ray theory Travel-times for head Model geometry;calculation of travel-times waves and reflections travel-time only.of reflected and refracted for plane or dippingphases (Program TXCURV). layered models; exact.

Ray Tracing (Program Two-dimensional models. Travel-times only:RAY2D). limited number of

phases; ignores P-Sconversions, guidedwaves and surfacewaves.

Synthetic Seismogram Modeling

Reflectivity (Program Wave-theoretical solution Restricted to I-D. . SYNCAL). to response of a layered models.

half-space. Includes P, SV.r9 guided phases and surface

waves. Exact. Includes Q.

34

U

.. .. . . . . . _. . , . .. " ...-- - . .. . ., . . .. . . . . .. .-. ...,., . , ., - - - '. _t .L ",, . . , . .. , . . . b . .

L"

Table 3 (cont'd).

Method Characteristics Lim ita tions

Disk Ray Theory (modified 2-D models, approximate Inexact amplitudes forRAY2D program). solution for P waves, can head waves and near-

include reflection coeffi- critical arrivals; ignorescients and Q. Computa- guided phases, P-S con-tionally efficient. versions and surface

waves.

Finite Difference Acoustic 2-D models, exact solu- Acoustic rather than(Program FDWVAC) tion for all wave types elastic. Large computer

in an acoustic media, time.Significantly faster thanFDWVEQ, but slower

than RAY2D synthetics.

Finite Difference Elastic 2-D models, exact solu- Extensive computer time(Program FDWVEQ) tion for all wave types and storage.

in elastic media. Can bemodified to include Q.

Table 4. Characteristics of methods of generating synthetic seismograms.

Method Characteristics

Geometric ray theory Analytic. Usefulness in tracing propagation path of waveenergy depends on complexity of structure. Invalid whenchange in seismic velocity or curvature of wave front islarge over a wavelength. Does not predict frequency-depend-ent effects.

Generalized ray theory Analytic. Spherical waves incident on planar interfaces.(Cagniard-de Hoop) Incorporates specified direct, reflected, critically re-

fracted, and Rayleigh waves. Integration in complex rayparameter plane and a Laplace transform. Vatidity ofsynthetic seismograms dependent on which rays are in-corporated in solution. Truncation error results whenfinite number of rays is used.

Asymptotic ray theory Analytic. Full solution is sum of infinite series of fre-quency-dependent terms. Asymptotic approximation, validat high frequencies, consists of I or 2 terms. Additionalterms necessary in the case of diffraction phenomena, curvedinterfaces, or inhomogeneous media. Increased accuracy ofbody-wave amplitudes obtained with various modificationsof method.

Reflectivity Analytic. Spherical waves propagating in plane-layeredmedium. Reflectivity term is costly to compute but needbe computed only once for a given model. All internalmultiples and conversions automatically included in re-flectivity term. Two integrations required. Integration r .over limited range of real ray parameters restricts solu-tion to body waves and introduces truncation error.

Full wave theory Analytic. Real earth models (radial inhomogeneity) andsmoothly varying velocity and density. Can incorporateray paths involving turning points, near-grazing incidence,and critical incidence; multipathing; diffraction phenomena;and caustics. Uses frequency-dependent reflection andtransmission coefficients. Two integrations required. Raysto be incorporated in solution must be specified.

Surface wave seismology Analytic. Highly sensitive to structure of layered medium.Results usefully presented as dispersion curves. Body wavesignored. Representation in terms of multiply reflected, in-terfering plane waves valid only if distance from layer inter- -

face to source is large relative to layer thickness. Valid at

35

-. - o - ,

Table 4 (cont'd). Characteristics of methods of generating synthetic seismograms.

Method Characteristics

Surface wave seismology frequencies that are too low to meet high frequency approxi-(cont'd) mation of ray/wave theory yet are high enough that required

number of normal modes is prohibitive.

Normal modes Analytic. Plane-layered waveguide handled readily. Validat all frequencies but practical at low frequencies (smallernumber of modes and so lower cost). Computational timereduced by employing Fast Field Program which can in-corporate both normal and leaky modes. Costly, perhapsinefficient, in case of complex structure because normalmodes must be recomputed whenever inhomogeneity inmodel is encountered.

Finite differences Numerical. Displacement at grid points. Useful in near-field or with complex structure. Body waves or surfacewaves. Size of grid spacing affects stability and accuracyof results. Truncation error due to approximation towave equation. Aliasing, numeric dispersion, and spuriousreflected waves from artificial boundaries are potentialproblems. Costly due to number of computations andstorage requirements.

Finite elements Numerical. Displacements between grid points known byinterpolation. Useful with irregularly shaped structure.Body waves or surface waves. Accuracy dependent oni-.mber of elements and method of interpolation. Alias-ing, numeric dispersion, and spurious reflected waves

" from artificial boundaries are potential problems. Costlydue to number of computations and storage requirements.

By presenting the characteristic features of each method, this report may overemphasize differ-ences and unintentionally obscure similarities among the methods. It should be realized that theaccuracy of synthetic seismograms generated by any of these methods is highly frequency depend-

- - ent. Frequency (f) and equivalently wavelength (N = V/f) serve as scale factors for a problem. Thevalidity of the synthetic seismogram obtained is constrained by grid spacing relative to wavelength

• "with numerical methods and by layer thickness relative to wavelength or change in velocity gradi-ent over a wavelength in analytic approaches to solving the wave equation. The condition for thevalid use of asymptotic approximations to replace exact quantities in the analytic solutions is gen-erally kr = 2rr/X >> 1, where r is the range from the source. In the far field, r is sufficiently large(at least 10 X) that the condition is satisfied. The actual propagation distance required for far-fieldconditions to prevail is inversely proportional to frequency.

An understanding of the limitations and approximations inherent to the methods of generatingsynthetic seismograms, together with the selection of a method appropriate to a particular problem,is necessary if synthetic seismograms are to be used wisely and advantageously.

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._% .. '-:? ' .*._ ;- -

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... ... . -- . . ...- _

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The Royal Astronomical Society, 37: 423-433. --

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A facsimile catalog card in Library of Congress MARCformat is reproduced below.

Peck, LindamaeReview of methods for generating synthetic seismo-

grams / by Lindamae Peck. Hanover, N.H.: U.S. ArmyCold Regions Research and Engineering Laboratory;Springfield, Va.: available from National TechnicalInformation Service, 1985.

v, 48 p., illus.; 28 cm. (CRREL Report 85-10.)Prepared for Office of the Chief of Engineers by

Corps of Engineers, U.S. Army Cold Regions Researchand Engineering Laboratory under DA Project 4A161102-AT24.

Bibliography: p. 36.1. Geophysics. 2. Ground motion. 3. Mathematical

analysis. 4. Seismic signatures. 5. Seismology. 6.Synthetic seismograms. I. United States. Army. Corpsof Engineers. II. Cold Regions Research and Engineer- Sing Laboratory, Hanover, N.H. V. Series: CRREL Report85-10.

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